Iran University of Science and Technology, Tehran

Iran University of Science and Technology, Tehran, Narmak-16, I.R. Iran. January, 29-30, 2014

Contents 1 Full Papers 1 1.1 Z. Bakefayat :: Oscillator topologies on a paratological group . . . . . . . . 2 1.2 Z. Aral and A. Razavi :: Evolution of the curvature under the Finsler Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 M. Nadjafikhah and L. Hamedi Mobarra :: The Milne metric and reductions of its Gordon- type equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 M. Deldar Froutagheh :: Free paratopoligical groups . . . . . . . . . . . . . 17 1.5 M. Parhizkar and P. Bahmandoust and D. Latifi :: Bi-invariant Finsler metrics on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 E. Keyhani :: Generalized differentiation in modules over topological ∗algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.7 A. Babaee, B. Mashayekhy and H. Mirebrahimi :: On Hawaiian groups of pointed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8 B. Bidabad and M. Yarahmadi :: Ricci flow and convergence of evolving Finslerian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.9 M. Nadjafikhah and M. Hesamiarshad :: The equivalence problem for equations of the form uxxx = ut + A(t, x)ux + B(t, x)u + C(t, x) . . . . . . . . . 44 1.10 B. Bidabad and M. Sedaghat :: Extrinsic sphere in Finsler geometry . . . . 50 1.11 M. Nadjafikhah and P. Kabi-Nejad :: Conservation Laws of the Kupershmidt equation by the Scaling method . . . . . . . . . . . . . . . . . . . . . . 56 1.12 Z. Jamal Kashani and B. Bazigaran :: Order-representability of topological spaces and preorderable and lower preorderable topologies . . . . . . . . . . 62 1.13 B. Najafi, F. Malek and N. H. Kashani :: Contact Finsler structure . . . . . 67 1.14 M. Nadjafikhah and K. Goodarzi :: Integrating factor and λ-symmetry for third-order differential equations . . . . . . . . . . . . . . . . . . . . . . . . 73 1.15 R. Mirzaie and H. Soruosh :: A remark on topology of some UND-manifolds 82 1.16 M. Faghfouri and R. Hosseinoghli :: More on Warped product Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.17 M. Nadjafikah, A. Mahdavi and M. Toomanian :: Two approaches to the calculation of approximate symmetry of a Ostrovsky equation with small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 1.18 U. Mohamadi :: Dynamic topological entropy of co-compact open covers . . 99 1.19 D. Latifi and M. Toomanian :: On reduced Finsler Σ-spaces . . . . . . . . . 105 1.20 M. Nadjafikah and H. Reza Yazdani :: Apply Fushchych method on celebrated version of Fisher-KPP equation . . . . . . . . . . . . . . . . . . . . . 110 1.21 A. Pakdaman :: On the Mycielski conjecture . . . . . . . . . . . . . . . . . 116 1.22 E. Pouranvari and G. Haghighatdoost :: The topological features for a new case on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5

1.23 F. Ahangari :: Comprehensive geometric investigation of Finslerian integrable cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.24 R. Bakhshandeh Chamazkoti :: The gauge equivalence of fourth order differential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.25 N. Abbasi and M. Pourbarat :: A dynamic criteria for recognizing the lacking Baire property for some topological spaces . . . . . . . . . . . . . 1.26 M. Zohrehvand and M. Mirmohamah Rezaii :: On the H-curvature of Einstein (α, β)-metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.27 M. Sabzevari :: Cartan equivalence problem of the 5-dimensional universal CR-model Mc5 ⊂ C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28 M. Amini :: Smooth vectors and integral curves of lie hypergroups . . . . 1.29 M. Nadjafikhah and M. Yaghesh λ-symmetries and solution of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.30 R. Chavosh Khatamy and U. Ghalebsaz Jedy :: Construction of invariant (α, β)-metrics on reductive homogeneous spaces . . . . . . . . . . . . . . 1.31 V. Shirvani-Sh, M. Nadjafikhah and M. Toomanian :: Application of Lie symmetries to construct conservation laws of PDEs . . . . . . . . . . . . . 1.32 H. Torabi, A. Pakdaman and B. Mashayekhy :: On the Classification of Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33 Y. Alipour Fakhri and S. Loghmannia :: The horizontal Hodge operator in Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.34 M. Nadjafikhah and Z. Pahlevani Tehrani :: Solving equivalence problems with equivariant moving frame method under finite dimentional Lie group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.35 M. Rabii :: The combinatorial model of the multibrot set for d = 4 . . . 1.36 F. Ayatollah Zadeh Shirazi and B. Taherkhani :: A short note on ergodicity: in the class of topological transformation semigroups with phase semigroup as a collection of generalized shifts . . . . . . . . . . . . . . . . . . . . . . 1.37 R. Akbarzadeh and G. Haghighatdoot :: Investigation of the topological structure and the loop molecule of the saddle-saddle critical point of complexity of 4 on some integrable Hamiltonian system . . . . . . . . . . . . . 1.38 F. Douroudian :: Combinatorial knot Floer homology and double branched covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.39 A. Etemad :: Surfaces in Euclidean spaces of higher dimensions . . . . . . 1.40 A. Mohammadpouri :: On r-minimal isometric immersions from warped products into Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . 1.41 S.E. Akrami :: Quantum (fuzzy or none pure) differential geometry . . . . 1.42 F. Ayatollah Zadeh Shirazi and M.A. Mahmoodi :: which functional Aexandroff topological spaces are COTS? . . . . . . . . . . . . . . . . . . . . . . 1.43 B. Raesi and F. Bakhtiari :: Geometrical Approach of bifurcations in periodically forced Morris Lecar neurons . . . . . . . . . . . . . . . . . . . . . 1.44 M. Nadjafikhah and M. Korshidi :: Lie symmetries of the fractional-partial diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.45 L. Zareh Yazdeli and B. Bazigaran :: Cut point in topological spaces . . . 1.46 B. Raesi and K. Horr :: Geometric approach of bursting in a hyperchaotic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.47 M.B. Kazemi :: Integrable distributions on semi-invariant submanifolds . 1.48 R. Chavosh Khatamy and D. Latifi :: On symmetry preserving diffeomorphisms of generalized symmetric Finsler spaces . . . . . . . . . . . . . . .

. 130 . 135 . 143 . 148 . 156 . 162 . 171 . 174 . 179 . 184 . 190

. 194 . 200

. 206

. 212 . 219 . 224 . 230 . 234 . 246 . 252 . 258 . 263 . 267 . 273 . 277

1.49 M. Parhizkar, D. Latifi and P. Bahmandoust :: Sectional curvature and Scalar curvature two-step Nilpotent Lie groups of dimension five . . . . . . 283 1.50 M. Aghasi and M. Nasehi :: (κ, µ)-Spaces which are isometrically immersed In an almost Hermitian kahler manifold . . . . . . . . . . . . . . . . . . . . 289 1.51 F. Ayatollah Zadeh Shirazi and R. Rahimi :: Interaction between relatively pointwise recurrence and dense periodic points in two class of topological dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 1.52 G. Haghighatdoost, R. Mahjoubi and R. Akbarzadeh :: New integrable Hamiltonian systems on the Lie algebra e(3 ) and topology of their isoenergetic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 1.53 J. Abedi-Fardad, G. Haghighatdoost and A. Rezaei-Aghdam :: Integrable and superintegrable Hamiltonian systems with four dimensional real Lie algebras as symmetry of the systems . . . . . . . . . . . . . . . . . . . . . . 307 1.54 A. Haji Badali and M. Asadollahzadeh and E. Sourchi :: Sasakian and Conformal Contact Lorentz Manifold . . . . . . . . . . . . . . . . . . . . . . 313 1.55 H. Sahleh and Hossein E. Koshkoshi :: On first non-abelian cohomology of topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 1.56 R. Abdi and E. Abedi :: Totally umbilic CR-hypersurfaces of a Kenmotsu space form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 1.57 A. Haji Badali and M. Asadollahzadeh and F. Alizadeh :: A note on contact Lorentz manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 1.58 M. Asadollahzadeh, A. Haji Badali and E. Azimpour :: A Study on Contact Pseudo-Metric Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 1.59 M. Asadollahzadeh and A. Haji Badali and R. Karami :: Certain Contact CR-submanifolds of a Lorentzian Sasakian Manifold . . . . . . . . . . . . . 341 1.60 A. Hosseini and H. Sahleh :: Generalization of topological local group . . . 347 1.61 M. Jafari :: Classical and nonclassical symmetries of the 2-dimensional Ricci flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 1.62 F. Ayatollah Zadeh Shirazi :: Co-decomposability of a transformation semigroup to non-proximal transformation semigroups . . . . . . . . . . . . . . . 358 1.63 M. Faghfouri and T. Kasbi :: Minimal translation surfaces in Sol3 with the Lorentz metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 1.64 F. Heydari and D. Behmardi :: Fragmentability of topological spaces . . . . 370 1.65 T. Akbarzade and H. Haghighi :: Higher secant varieties of parameterizing varieties of the variety of completely decomposable forms . . . . . . . . . . 376 1.66 P. Ahmadi :: On cohomogeneity one actions of a Heisenberg Lie group on some Lorentz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 1.67 G.A. Haghighat Doost, R. Mahjobi and H. Abedi Karimi :: Generalization of the compatible Lie-Poisson brackets on the Lie algebra so(4) . . . . . . . 386 1.68 N. Elyasi and N. Broojerdian :: Application of Lie algebroid structure in unification of field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 1.69 A. Zaeim and M. Chaichi :: On Lorentzian four-manifolds with special kind of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 1.70 F. Akhtari and R. Nasr-Isfahani :: Continuous involutions on group algebras with a large family of topologies . . . . . . . . . . . . . . . . . . . . . . . . 405 1.71 F. Pashaie :: On k-stable hypersurfaces in Riemannian and Lorentzian space foroms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 1.72 N. Boroojerdian and M. Imanfar :: Hamiltonian mechanics on Lie algebroids414

1.73 M. Mir Mohammad Rezaii and V. Pirhadi :: Lie algebroid on contact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 1.74 S. Azami and A. Razavi :: Eigenvalues variation of the Laplace operator under the Yamabe flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 1.75 B. Raesi and F. Arab :: Morris-Lecar neuronal networks topology and synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 1.76 B. Bidabad and A. Shahi :: Estimates on covariant derivatives of cartan HH-curvature under Ricci flow . . . . . . . . . . . . . . . . . . . . . . . . . 436 1.77 M. Nadjafikhah :: Group analysis of three dimensional Euler equation of gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 1.78 M. Nadjafikhah, A. Mahdipour Sh., R. Bakhshandeh Chamazkoti :: Galilean classification of spacetime curves . . . . . . . . . . . . . . . . . . . . . . . . 451 1.79 M. Nadjafikhah :: Exact solution of generalized inviscid Burgers’ equation . 457 1.80 M. Nadjafikhah and S.R. Hejazi :: Symmetries of 2nd and 3rd order homogeneous ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 1.81 M. Nadjafikhah :: Classification of n−th order linear ODEs up to projective transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 1.82 F. Ayatollah Zadeh Shirazi and Z. Nili Ahmadabadi :: Finally distality . . 473 1.83 F. Malek and M. Mirghafouri :: A flow on Hessian manifolds . . . . . . . . 479 1.84 B. Bidabad :: Finsler geometry and some recent developments . . . . . . . . 485 1.85 E. Abedi, M. Asadollahzadeh and Gh. Haghighatdoost :: Codimension reduction on contact CR-submanifold of an odd dimensional unit sphere . . 493 1.86 M. Abry, A. Forouzandeh :: Hyperspaces with Hausdorff metric and ANR’s 499 1.87 L. Zareh Yazdeli and B. Bazigaran :: QHC space . . . . . . . . . . . . . . . 505 1.88 F. Rezaee Abharee :: The rees-suschkewitsch theorem for simple topological semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 1.89 H. Sahleh and A. Alijani :: S-pure extensions of locally compact Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 1.90 N. Mohammadi and Z. Nazari :: Golden structures of 4-dimensional connected metric lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 1.91 S. R. Hejazi and N. Kushki :: Two methods for finding symmetries of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 1.92 M. Ebrahimi and A. Mehrpooya :: An application of geometry in algebra: uncertainty of hyper mvalgebras . . . . . . . . . . . . . . . . . . . . . . . . 529 1.93 M. R. Farhangdoost and S. Kiyani :: Comparison of three generalized open sets ρ, ν, ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 1.94 B. Bazigaran and S. Baharlouie :: General box product topology . . . . . . 539 1.95 M. Montazeri and B. Bazigaran :: Jump in the order of the lattice of T1 -topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 1.96 M. Montazeri and B. Bazigaran :: Upper topologies in the lattice of T1 -topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 1.97 Z. Jamal Kashani and B. Bazigaran :: The existence of order monomorphism and order representability of topological spaces . . . . . . . . . . . . 562 1.98 F. Mottaghi :: Recursive determinants . . . . . . . . . . . . . . . . . . . . . 568 2 General Talks 2.1 B. Bidabad :: Finsler geometry and some recent developments . . . . . . . 2.2 F. Malek :: Chen inequality and its improvements . . . . . . . . . . . . . 2.3 H. Khorshidi :: Digital topology is not general topology . . . . . . . . . .

575 . 576 . 577 . 578

2.4 2.5

M. Nadjafikhah :: On the varitional bicomplex . . . . . . . . . . . . . . . . 579 A. Dehghan Nezhad :: Some recent results and questions concerning the cohomological equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

Chapter 1

Full Papers

1

Z. Bakefayat :: Oscillator topologies on a paratological group

Paper No. 1.1

Oscillator topologies on a paratological group 1

Zahra Bakefayat1 e-mail: [email protected]. Abstract

In this note we introduced oscillator topologies on paratopological group admits a weaker Hausdorff group topology. Keywords: n-oscillating protopological group, saturated paratopological group, ω -bounded, b-separated

1 Introduction Does every Hausdorff paratopological group G admit a weaker Hausdorff group topology? Under a paratopological group we understand a pair (G, τ ) consisting of a group G and a topology τ on G making the group operation : G × G −→ G of G continous. If, in addition, the operation (.)−1 : G −→ G of taking the inverse is continuous with respect to the topology τ , then (G, τ ) is a topological group. Given a paratopological group G let τb be the strongest group topology on G, weaker than the topology of G. The topological group Gb = (G, τb ) called the group reflexion of G, has the following characteristic property: the identity map i : G −→ Gb is continuous and for every continuous group homomorphism h : G −→ H from G into a topological group H the homomorphism hoi−1 : Gb −→ H is continuous. A subset A of a protopological group G will be called b-closed if A is closed in the topology τb . A protopological group G is called b-separated provided its group reflexion Gb is Hausdorff. For a n−1 −1 −1 −1 (−1)n subset U of a group G we define that (±U)n = UU U ·{z · ·U (−1) } and (∓U)n = U | | UU {z· · ·U } n

n

and (±U)0 = (±U)0 = e. Note that ((±U)n )−1 = (±U)n if n is even and ((±U)n )−1 = (∓U)n if n is odd.

2 Oscillator topologies on a paratopological group and saturated paratopological group Under an n-oscillator on a topological group (G, τ ) we understand a set of the form (±U)n for some neighborhood U of the unit of G.

2


Under the n-oscillator topology on a paratopological group (G, τ ) we understand the topology τn consisting of sets U ⊂ G such that for each x ∈ U there is an n-oscillator (±V )n with x.(±V )n ⊂ U. For a paratopological group (G, τ ) with finite oscillation, osc(G) be the smallest n ∈ W such that for any neighborhood U ⊆ G of e the set (±U)n is a neighborhood of e in Gb . We say that a paratopological group (G, τ ) is n-oscillating if osc(G) ≤ n. In particular, a paratopological group G is 3-oscillating (resp. 2-oscillating) provided for any neighborhood U of the unity e of G there is a neighborhood V ⊂ G of e such that V −1VV −1 ⊂ UU −1U (res. V −1V ⊂ UU −1 ). The paratopological group (G, τ ) called saturated if for any neighborhood U ⊂ G of e, the set U −1 has nonempty interior in (G, τ ). for example the Sorgenfrey line is a saturated paratopological group. Suppose G is a saturated paratopological group, then G is 2-oscillating. The following proposition can be easily derived from the definitions and the equality (τn )−1 = (τ −1 )n holding for each odd n. Suppose G is a topological group with finite oscillation and G− is its mirror paratopological group. Then 1. If osc(G) is odd, then osc(G) − 1 ≤ osc(G− ) ≤ osc(G). 2. If osc(G) is even, then osc(G) ≤ osc(G− ) ≤ osc(G) + 1. It is clear that each 2-oscillating paratopological group is 3-oscillating. We shall show that 3-oscillating paratopological groups are b-separated. Any Hausdorff 3-oscillating paratopological group (G, τ ) is b-separated. A Hausdorff paratopological group G is b-separated, if G is a saturated paratopological group. The paratopological group (G, τ ) called ω -bounded if for any U ∈ ζ (e) (ζ (e) denotes the family of neighborhood of the identity e.) there is a countable subset that G = FU = UF. Here the relevance of saturated paratopological groups lies in that saturated Hausdorff paratopological groups admit a weaker Hausdorff group topology. The following result says that we can obtain saturated paratopological group by combining the properties ω -bounded and Baire space. Every ω -bounded Baire paratopological group (G, τ ) is saturated. 2-space (X, τ , σ ) is 2-pseudocompact if f : (X, τ , σ ) −→ (R,U, L) is bounded in R. (U, L are respectively upper limit topology and lower limit topology on R). Since every 2-pseudocompact paratopological group is a Baire space and Theorem 2 we obtain: Every ω -bounded 2-pseudocompact Hausdorff paratopological group admits a weaker Hausdorff group topology.

3


The previous result provides a wide class of 2-pseudocompact Hausdorff paratopological groups which admits a weaker Hausdorff group topology: for instance, Lindelof or separable 2-pseudocompact (in particular countably compact) Hausdorff paratopological groups.

References [1] O. T. A LAS AND M. S ANCHIS, Countably compact paratopological groups, Semigroup Forum 74 (2007), 423-438. [2] T. O. BANAKH AND O. R EVSKY, Oscillator topologies on a paratopological group and related number invariants, Submitted to a special issue debicated to the memory of M. Ostrogradesky published by Kyiv Institue of Mathematics.

4

Z. Aral and A. Razavi :: Evolution of the curvature under the Finsler Ricci flow

Paper No. 1.2

Evolution of the curvature under the Finsler Ricci flow 1

Zohreh Aral1 and Asadollah Razavi 2 e-mail: [email protected], Shahid Bahonar University of Kerman, Kerman, Iran. 2 e-mail: [email protected], Shahid Bahonar University of Kerman, Kerman, Iran Abstract

∂ We consider the Finsler Ricci flow F 2 = −2F 2 Ric, when the Finsler metric evolves, then ∂t so does its curvature. In this paper, we drive evolution equations for Ricci scalar function and the curvature tensors along the Finsler Ricci flow. We further show that Ricci scalar is nonnegative if it is nonnegative at t = 0. Keywords: Ricci flow, Finsler manifold, Berwald metric, curvature.

1 Introduction ∂ The geometric evolution equation g = −2Ric is known as the un-normalized Ricci flow in Riemannian ∂t geometry. The same equation can be used in the Finsler setting, because both the fundamental tensor gi j and Ricci tensor Rici j have been generalized to that broader framework, albeit gaining a y dependence in the process. Bao, in [1], studied Ricci flow equation in Finsler manifolds, whereby to answer Chern’s question that, where every smooth manifold admits a Ricci- constant Finsler metric? In the following a scalar Ricci flow equation is introduced according to the Bao’s paper. By contracting ∂ 1 gi j = −2Rici j = −2( F 2 Ric)yi y j ∂t 2 with yi and y j gives, via Euler’s theorem,

∂ F2 = −2F 2 Ric, that is ∂t

∂ logF = −Ric, F(t = 0) = F0 ∂t This scalar equation directly addresses the evolution of the Finsler metric F, and makes geometrical sense on both the manifold of non zero tangent vectors T M0 and the manifold of rays. It is suitable as an un-normalized Ricci flow for Finsler manifold. In [2] Azami and Razavi showed that the Ricci flow on Finsler manifolds with Berwald metrics cannot possibly be strictly parabolic then, they defined a manifold flow which is strictly parabolic and using it, they proved the existence and uniqueness for solution of Ricci flow on Finsler manifolds.

5


2 Preliminaries Let M be a connected n-dimensional smooth manifold.Denote by Tx M the tangent space at x ∈ M,and by T M = ∪x∈M Tx M the tangent bundle of M.Any element of T M has the form (x,y), where x ∈ M and y ∈ Tx M.The natural projection π : T M → M is given by π (x,y) = x.Denote the pull-back tangent bundle by π ∗ T M which is π ∗ T M = {(x,y,v)|y ∈ Tx M0 ,v ∈ Tx M} and T M0 = T M\{0}, π (v) = x. A Finsler metric on a manifold M is a function F : T M → [0,∞) which has the following properties: (i) F(x, λ y) = λ F(x,y), ∀λ > 0; (ii) F(x,y) is C∞ on T M0 ; (iii) For any tangent vector y ∈ Tx M,the symmetric bilinear form gy : Tx M × Tx M → R on T M is positive definite,where ] 1 ∂2 [ 2 gy (u,v) := F (x,y + su + rv) . 2 ∂ s∂ r s=r=0

We call the indicatrix of the Finsler manifold (M,F) at the point x ∈ M, the set of unit tangent vectors ( ) 2 2 SM = {y ∈ Tx M : F(x,y) = 1}.In a local coordinate system xi ,yi we have gi j (x,y) = 12 ∂∂yi ∂Fy j (x,y) and (gi j ) := (gi j )−1 .The pair (M,F) is called a Finsler manifold.The geodesics of F are characterized locally by dx d 2 xi + 2Gi (x, ) = 0 2 dt dt where } { 1 il ∂ g jl ∂ g jk j k i G= g 2 k− l yy (2.1) 4 ∂x ∂x A Finsler metric is called a Berwald metric if the geodesic coefficients Gi (y) are quadratic in y ∈ Tx M for all x ∈ M,that is,there are local functions Γijk (x) on M such that Γijk (x) = Γik j (x) and

where

1 Gi (y) = Γijk (x)y j yk 2

(2.2)

∂ g jk 1 ∂ gl j ∂ g jk ∂ gkl ∂ gl j ∂ gkl Γijk = gil { k − l + j − r Grk + r Grl − r Grj } 2 ∂x ∂y ∂y ∂y ∂x ∂x

(2.3)

and Gij = ∂∂ Gy j . For a Berwald metric we have Γijk = ∂∂y j ∂Gyk . For a vector field y ∈ Tx M0 ,the Berwald connection is a map ∇y : Tx M ×C∞ (T M) → Tx M defined by { ( ) } ∂ y i j i k ∇uV := u V (x) +V (x)Γ jk (y)u (2.4) ∂ xi x i

2 i

6


where u = ui ∂∂xi ∈ Tx M and V = V i ∂∂xi ∈ C∞ (T M).From now on a vector field y ∈ Tx M0 x that ∇ = ∇y . The coefficients of the Riemann curvature Ry = Rik dxi ⊗ ∂∂xi are given by Rik := 2 ( and Ri

jk

:=

1 3

∂ Ri j ∂ yk

∂ Ri − ∂ y jk

2 i ∂ Gi ∂ 2 Gi j ∂ Gi ∂ G j j ∂ G − y + 2G − ∂ x k ∂ x j ∂ yk ∂ y j ∂ yk ∂ y j ∂ yk

we suppose

(2.5)

) ( 2 i ) ∂ R ∂ 2 Ri , R ij kl := 31 ∂ y j ∂ yk l − ∂ y j ∂ yl k .

The Ricci scalar function of F is given by Ric :=

1 i R. F2 i

A companion of the Ricci scalar is the Ricci tensor ( ) 1 2 . Rici j := F Ric 2 yi y j

(2.6)

(2.7)

A Finsler metric is said to be an Einstain metric if the Ricci scalar function is a function of x alone,equivalently Rici j = R(x)gi j .

3 Evolution of the curvature under the Finsler Ricci flow ∂ F2 Theorem 3.1. Under the Finsler Ricci flow, = −2F 2 Ric, the evolution of the (1,3) curvature tensor ∂t is given by ∂ i (R ) = ∂ H ∗ ∂ g + ∂ g ∗ ∂ h + H ∗ ∂ 2 g + g ∗ ∂ 2 h + (g ∗ ∂ g)(H ∗ ∂ g + g ∗ ∂ h) ∂ t jkl ∂ ij g . ∂t Proof: We compute where H i j =

1∂ ∂ i ∂ 3 Gi (R jkl ) = ( l j k ) + (l ←→ k) + 2( j ←→ l) ∂t 3 ∂t ∂x ∂y ∂y 1 ∂ ∂ 2 Gm ∂ 2 Gi 2 1 + ( )( m k ) − ( j ←→ k) + (l ←→ k) j l 3 ∂t ∂y ∂y ∂y ∂y 3 3 2 m 2 i 1 ∂ G ∂ ∂ G 2 1 + ( j l )( ) − ( j ←→ k) + (l ←→ k) m k 3 ∂y ∂y ∂t ∂y ∂y 3 3 = I + II + III

7

(3.1)


we have

∂ i 1 ∂ H ic ∂ g jc ∂ g jk ∂ gkc ∂ gk j ∂ gic ∂ h jc ∂ h jk ∂ hkc ∂ hk j (R jkl ) = [ j (2 k − c + 2 j − c ) + l (2 k − c + 2 j − c ) ∂t 12 ∂ x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 2 2 2 ∂ g jk ∂ gk j ∂ g jc ∂ gkc + H ic (2 l k − l c + 2 l j − l c ) ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2h 2 ∂ ∂ 2 hk j ∂ 2 hkc jk ic ∂ h jc + g (2 l k − l c + 2 l j − l c )] + (l ←→ k) + 2( j ←→ l) ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x The second and third terms are 1 mc ∂ g jc ∂ glc ∂ gl j ic ∂ gmc ∂ gkc ∂ gkm ∂ hmc ∂ hkc ∂ hkm II = g ( l + j − c )(h ( k + m − c ) + gic ( k + m − c )) 12 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x + (k ←→ j) + (l ←→ k) I=

∂ h jc ∂ hlc ∂ hl j 1 ic ∂ gmc ∂ gkc ∂ gkm mc ∂ g jc ∂ glc ∂ gl j g ( k + m − c )(h ( l + j − c ) + gmc ( l + j − c )) 12 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x + (k ←→ j) + (l ←→ k)

III =

We can rewrite the formula as I = ∂ H ∗ ∂ g + ∂ g ∗ ∂ h + H ∗ ∂ 2g + g ∗ ∂ 2h II + III = (g ∗ ∂ g)(h ∗ ∂ g + g ∗ ∂ h). □ Theorem 3.2. Suppose g(t) is a smooth one-parameter family of Berwald metrics on a manifold M such ∂ F2 that = −2F 2 Ric, then the evolution of curvature tensor evolves by ∂t ∂ i R = ∂ H ∗ (∂ g)y + ∂ g ∗ (∂ Ric)y + H ∗ (∂ 2 g)y + g ∗ (∂ 2 Ric)y ∂ t jk + (H ∗ (∂ g)y + g ∗ (∂ Ric)y)(g ∗ ∂ g) + (g ∗ (∂ g)y)(H ∗ ∂ g + g ∗ ∂ Ric) Proof: We compute 1 ∂ ∂ 2 Gi ∂ i ∂ ∂ Gm ∂ 2 Gi ∂ Gm ∂ ∂ 2 Gi R jk = ) + ( ) − ( j ←→ k) + ( )( )( ∂t 3 ∂ t ∂ yk ∂ x j ∂ t ∂ yk ∂ ym ∂ y j ∂ yk ∂ t ∂ ym ∂ y j = I + II + III we have 1 ∂ H il ∂ gkl ∂ gkb b ∂ gal ∂ gak a 1 ∂ gil ∂ hkl ∂ hkb b ∂ hal ∂ hak a [(2 − y + (2 − )y ] + [(2 − y + (2 − )y ] 12 ∂ x j ∂ xb ∂ xl 12 ∂ x j ∂ xb ∂ xl ∂ xk ∂ xl ∂ xk ∂ xl ∂ 2 gkb ∂ 2 gal ∂ 2 gak 1 il ∂ 2 gkl + H [(2 b j − l j )yb + (2 k j − l j )yb ] 12 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 2 2 1 il ∂ hkl ∂ hkb ∂ 2 hal ∂ 2 hak + g [(2 b j − l j )yb + (2 k j − l j )yb ] − (k ←→ j) 12 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x

I =

8


The second term is 1 1 ∂ gkl ∂ gkb ∂ gal ∂ gak ∂ hkl ∂ hkb ∂ hal ∂ hak II = ( H ml [(2 b − l )yb + (2 k − l )ya ] + gml [(2 b − l )yb + (2 k − l )ya ]) 4 4 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 1 ∂ gmc ∂ g jc ∂ g jm ( gic ( j + m − c )) − (k ←→ j) 2 ∂x ∂x ∂x The third term is 1 ∂ gkl ∂ gkb ∂ gal ∂ gak 1 ∂ gmc ∂ g jc ∂ g jm 1 ∂ hmc ∂ h jc ∂ h jm III = ( gml [(2 b − l )yb + (2 k − l )ya ])( H ic ( j + m − c ) + gic ( j + m − c )) 4 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x − ( j ←→ k) We can rewrite the formulas as I = ∂ H ∗ (∂ g)y + ∂ g ∗ (∂ Ric)y + H ∗ (∂ 2 g)y + g ∗ (∂ 2 Ric)y II = (H ∗ (∂ g)y + g ∗ (∂ Ric)y)(g ∗ ∂ g) III = (g ∗ (∂ g)y)(H ∗ ∂ g + g ∗ ∂ Ric). □ Proposition 3.3. Under the Finsler Ricci Flow the evolution of the scaler curvature is given by

∂ ∂ 2 Ricrl ∂ 2 Ricrk ∂ 2 Ricsl ∂ 2 Ricsk 1 Ric = 2Ric2 + 2 gsl (−2 s k + s l + r k − r l )yr yk ∂t F ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x + lower order terms. Proof: Using the Finsler Ricci flow and (2.6), we have

∂ 1 ∂ Ric = 2Ric2 + 2 ( Rkk ) ∂t F ∂t by (2.5) we compute the time derivative of Rkk as,

∂ k 1 sl ∂ 2 hrl ∂ 2 hrk ∂ 2 hsl ∂ 2 hsk Rk = g (2 s k − s l − r k + r l )yr yk + lower order terms. ∂t 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x □

3.1 Controlling the Ricci curvature Conclusion: To go beyond controlling the scalar curvature, it is natural to consider a generalisation of the maximum principle which applies to symmetric 2-tensors. This was done by Hamilton in his paper on three manifolds [6]. In order to state his result, we need the following definition.

9


Definition 3.4. [3](Null-eigenvector assumption) We say β : Sym2 T ∗ M × [0,T ) −→ Sym2 T ∗ M satisfies the null-eigenvector assumption if whenever ωi j is a nonnegative symmetric 2-tensor at a point x, and if V ∈ Tx M is such that ωi jV j = 0, then βi j (ω ,t)V iV j ≥ 0 for any t ∈ [0,T ). Note that a symmetric tensor ωi j is defined to be non-negative if and only if ωi j vi v j ≥ 0 for all vectors vi (i. e. if the quadratic form induced by ωi j is positive semi-definite). In this situation we write ωi j ≥ 0. ∂ Remark 3.5. Let (M,F,d µ ) be a Finsler m space, d µ = σ (x)dx1 ...dxn and ∇ f = ∇i f i . The Laplacian ∂x 1 ∂ i (σ ∇ f ). of f is expressed by ∆ f = σ ∂ xi We naturally want to apply the maximum principle for tensor equations (see [5], page 97) to the evolution of Ricci scalar under the Finsler Ricci flow which by proposition (3.3) takes the form

∂ Ric ≥ ∆Ric + β (Ric,t) ∂t 1 ∂ where β = 2 ( Rkk )−∆Ric is completely expressed in terms of the Ricci scalar and its time derivative. F ∂t With the equation now in this form, one can simply check the null-eigenvector condition with X = 0, α = Ric and β to prove the following: Theorem 3.6. Suppose g(t), t ∈ [0,T ) is a solution of the Finsler Ricci flow on a compact Finslerian manifold (M,F), where g(t) is a Berwald metric. If Ric ≥ 0 at t = 0, then Ric ≥ 0 on 0 ≤ t < T .

References [1] D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics XX, 2007. [2] S. A ZAMI , A. R AZAVI, Existence and uniqueness for solution of Ricci flow on Finsler manifolds, International journal of geometric methods in modern physics, vol. 10, No. 03, 2013. [3] C. H OPPER , B. A NDREWS, The Ricci flow in Riemannian geometry, springer, 2010 . [4] Z. S HEN, Differential geometry of spray and finsler spaces,kluwer academic publishers, 2001. [5] B. C HOW, D. K NOPF, The Ricci flow an introduction, American mathematical society, 2004 . [6] R. S. H AMILTON, Three-manifolds with positive Ricci curvature, Journal of Differential geometry, vol. 17, No. 2, 255-306, 1982.

10

M. Nadjafikhah and L. Hamedi Mobarra :: The Milne metric and reductions of its Gordon- type equation

Paper No. 1.3

THE MILNE METRIC AND REDUCTIONS OF ITS GORDON- TYPE EQUATION Mehdi Nadjafikhah1 and Leila Hamedi Mobarra2,* e-mail: m [email protected], Iran University of Science and Technology, Tehran, Iran. 2 e-mail: [email protected], Science Department, Payamnoor University , 19395-4697, Tehran. I.R. Iran. * Department Of Science, Fouman And Shaft Branch, Islamic Azad University, Fouman, Iran. 1

Abstract The Milne metric, which has been proposed by Edward Arthur Milne to explain a model for an expanding empty space-time; satisfies a relation that is named the Gordon- type equation. The study of symmetries and Lie reductions of this equation has been done, earlier than this (according to [1]). This work devotes to introduce of both the Milne metric and the new findings of its Lie reductions and similarity solutions. Keywords: Milne metric, Gordon equation, Lie reduction, Similarity solutions.

1 Introduction space-time is a mathematical model for universe; as a single continuum, which has been produced by combining three dimensions of space and one dimension of time. The first form of the model was the consequence of the special theory of relativity; that was found by Albert Einstein in 1905, then was completed by Hermann Minkowski in 1908. This model, which is named 4-space Minkowski or cone isotropic or cone light; is a four dimensional flat manifold; that is endowed with the following Lorentzian metric ds2 = −dt 2 + dx2 + dy2 + dz2 . (1.1) The emersion of Einstein’s general relativity theory in 1916, after that the Hubble law in 1929, which respectively assumed that the world has curvature and expansion; led mathematicians to introduce a more complete space-time form. In this model, space-time is a four dimensional homogeneous and isotropic Lorentzian manifold; such that since the moment of the Big Bang has been expanding in the line of the Hubble flow (ref to [2, 3]). The metric of this space was innovated by Friedmann, Lematre, Robertson and Walker. This metric, that is named the FLRW or FRW- metric, is as the following ds2 = −dt 2 +

s2 (t) 2 (1 + k r4 )2

11

(dx2 + dy2 + dz2 ),

(1.2)


where r2 = x2 + y2 + z2 , and k = −1,0,1 is curvature constant (on the base of the Euclidean metric). Also, t is elapsed time since the Big Bang; t = 0 is the Big Bang moment, and p = (t,x,y,z) is comoving coordinate 1 of the point p of the universe (ref to [4]). With substitution k = −1 and s(t) = t in the FLRWmetric, the Milne metric is gained as 2 ds2 = −dt 2 +

t2 2 (1 − r4 )2

(dx2 + dy2 + dz2 ).

(1.3)

In this article, as we will show, the Milne metric has been considered in a conformal deformed 3 form. Some relations known as Gordon equations, are proposed to space-time field metrics (ref to [2]) . According to [1], the general form of Gordon-type equation corresponding to the Milne metric [gi j ] is 1 ∂ √ ∂ k(u) = √ ( | −g |gi j j u), (1.4) i ∂x | −g | ∂ x Where u = u(t,x,y,z), xi ∈ {t,x,y,z}; i = 1,··· ,4 and | −g |= det[−gi j ] , [gi j ] = [gi j ]−1 and k(u) is a non-negative integer power of u, or sin(u). In the next section, it will be noticed that (1.4) is a second order linear PDE. For this equation, related to k(u) = u3 and k(u) = u4 , some symmetries and reductions have been found, in 2011 (ref to [1]). In this work, we will research some Lie symmetries and reductions of the Gordon equation (1.4), for general k(u). As well as, we will acquire its similarity solutions; for k(u) = un ;n = 0,1,3.

2 The Milne metric and its Gordon equation The Milne space-time model is a special state of the FLRW model. This model is about an empty universe, and proposed by Edward Arthur Milne, in 1935 (ref to [4]). In the Milne model, space-time is accounted as a four dimensional flat expanding manifold (ref to [7]). The metric of this space is called the Milne metric. In the introduction, this metric has been presented in the form (1.3). It is notable that, in general, the Milne metric is introduced by ds2 = −dt 2 + t 2 ds2H 3 ; where ds2H 3 is a Riemannian metric on three dimensional hyperbolic H 3 (ref to [8]). In [1], The Milne metric (1.3) has been conformal deformed, and has been expressed in the form: ds2 = −dt 2 +t 2 (dx2 + e2x (dy2 + dz2 )). 1 comoving coordinate is the coordinate in which an observer is comoving with the Hubble flow.

(2.1)

In fact, comoving distance is distinct from actual distance. So, for two objects p1 and p2 in the cosmos, the actual distance is increasing. However, their comoving distance always remains constant (ref to [5]). 1 2 Where ds2 = (dx2 + dy2 + dz2 ) is the metric on three dimensional hyperbolic (ref to [6]). 2 2 H3 3 If [g

(1− r4 )

µν ] be a semi-Riemannian metric on (n + 1) dimensional M as ds2 = g00 (x)(dx0 )2 + ∑ni, j=1 gi j (x)dxi dx j ; x = {x j }, j = 1,··· ,n, Its conformal deformation is a metric like [g˜µν ], in the form g˜µν (x)= e2σ (x) gµν (x) ; σ (x) ∈ C∞ (M) (ref to [8]).

12


Theorem 2.1. The Gordon-type equation (1.4) related to the Milne metric is as ∆ = uxx −t 2 utt + e−2x (uyy + uzz ) − 3tut + 2ux −t 2 k(u) = 0,

(2.2)

Where k(u) ∈ {sin(u),un ;n = 0,1,2,···} . Remark 2.2. Per k(u) = u, (2.2) is named the Klein-Gordon equation.

3 Lie reductions of the Gordon equation related to the Milne metric 3.1

Lie point symmetries of the Gordon equation

In order to find the Lie point symmetry group, admitted by Eq. (2.2), take a general generator of symmetry algebra as

∂ ∂ ∂ + (a2 e−2x + ξ2 (t,x,y,z,u)) + (a3 e−2x + ξ3 (t,x,y,z,u)) ∂t ∂x ∂y ∂ ∂ +(a4 e−2x + ξ4 (t,x,y,z,u)) + (c1 e−2x + η1 (t,x,y,z,u)) ; (a1 ,··· ,a4 ,c1 ∈ R), (3.1) ∂z ∂u

X = (a1 e−2x + ξ1 (t,x,y,z,u))

Providing that Pr(2) X(∆) = 0. Performing the GroupActions package of the Maple, deduces that the symmetry Lie algebra is four-dimensional and is spanned by v1 = −

∂ ∂ ∂ ∂ ∂ ∂ ∂ + y + z , v2 = , v3 = −z + y , v4 = . ∂x ∂y ∂z ∂z ∂y ∂z ∂y

(3.2)

Theorem 3.1. The commutator relations of the symmetry Lie algebra are [v1 ,v2 ] = −v2 , [v1 ,v4 ] = −v4 , [v2 ,v3 ] = −v4 , [v3 ,v4 ] = −v2 .

(3.3)

On the other, the combining of the one parameter transformations exp(ε vi ) ; (i = 1,··· ,4) produces the Lie point symmetries group elements. Hence, it can be obtained: Theorem 3.2. The Lie point symmetries group of the Gordon equation (2.2), contains the following transformations (t,x,y,z,u) 7−→ (t,x − c1 ε ,ec1 ε (cos ε (y + c2 ) + sin ε (z + c3 )) + c4 , ec1 ε (cos ε (z + c5 ) + sin ε (y + c6 )) + c7 ,u). where ε , ci (i = 1,2,··· ,7) are arbitrary numbers.

13

(3.4)


3.2 Lie reductions and similarity solutions of the Gordon equation Theorem 3.3. One dimensional optimal system of symmetric subalgebras of the Gordon equation is conformity with (ref to [9]) ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ Θ1 = { v1 , v2 , v3 , v4 }. (3.5) Proof. Setting v = c1 v1 + ··· + c4 v4 ; (c1 ,··· ,c4 ∈ R), and using (3.3), for the Lie series Ad(exp(ε vi ))v = vi − ε [vi ,v] +

ε2 [vi ,[vi ,v]] − ··· ; (i = 1,··· ,4) 2

(3.6)

; necessitate Ad(exp(ε1 v1 ))oAd(exp(ε2 v2 ))oAd(exp(ε3 v3 ))oAd(exp(ε4 v4 ))v = c1 [e−ε1 (ε2 + (cos ε3 − sin ε3 ) ε4 ) + 1]v1 + c2 e−ε1 [cos ε3 + sin ε3 ]v2 + c3 [e−ε1 (−ε2 + (cos ε3 + sin ε3 )ε4 ) + 1]v3 +c4 e−ε1 [cos ε3 − sin ε3 ]v4 . (3.7) Any two brackets of the relation (3.7) do not equal to zero simultaneously. Furthermore, implementing of {ci ̸= 0, c j = 0; j ̸= i} in (3.7) for i = 1,2,3,4, results Θ1 ’s elements respectively. Therefore, the sentence is deduced.

Now, we (with ref to [9]) focus on reductions and their outcomes. I) For v1 = − ∂∂x + y ∂∂y + z ∂∂z :

dx By integrating of the characteristic equation dt0 = −1 = dyy = dzz = du0 , the corresponding invariants Y follow as: t,yex ,zex ,u. Substituting t = w, y = zZ Y , x = ln z , u = (t,x,y,z) = s(w,Y,Z), reduces eq.(2.2) to

(Z 2 + 1)sZZ + (Y 2 + 1)sYY + 2Y Z.sY Z + 3Y.sY + 3Z.sZ − w2 sww − 3wsw − w2 k(s) = 0.

(3.8)

The equation (3.8), do not has any symmetry for general k(s), or k(s) ̸= s. However for k(s) = s, it has an abnormal symmetry in Bessel functions expressions. II) For v2 = ∂∂z : By integrating of the characteristic equation dt0 = dx0 = dy0 = dz1 = du0 , the corresponding invariants follow as: t,x,y,u. Substituting t = w, x = Y, y = Z, u(t,x,y,z) = s(w,Y,Z), reduces eq.(2.2) to sYY − w2 sww + e−2Y sZZ − 3wsw + 2sY − w2 k(s) = 0. The equation (3.9), has two infinitesimal symmetries w1 = −Z ∂∂Z + ∂∂Y , w2 = ∂∂Z .

14

(3.9)


• For w1 : the corresponding invariants follow as: w, lnZ +Y, s. Substituting w = f , Y = g−lnZ, s(w,Y,Z) = T ( f ,g), reduces eq.(3.9) to − f 2 T f f + (1 + e−2g )Tgg − 3 f T f + (2 − e−2g )Tg − f 2 k(T ) = 0.

(3.10)

The equation (3.10), do not has any infinitesimal symmetry. • For w2 : the corresponding invariants follow as: w,Y,s. Substituting w = f , Y = g, s(w,Y,Z) = T ( f ,g), reduces eq.(3.9) to − f 2 T f f + Tgg − 3 f T f + 2Tg − f 2 k(T ) = 0. (3.11) The equation (3.11), has one infinitesimal symmetry u1 = ∂∂g . • For u1 : the corresponding invariants follow as: f , T . Substituting f = h, T ( f ,g) = S(h), reduces eq.(3.10) to hShh + 3Sh + hk(S) = 0.

(3.12)

Solving (3.12) for k(S) = 1, gives a similarity solution of eq.(2.2) via C1 1 S = − h2 + 2 +C2 . 8 h

(3.13)

Solving (3.12) for k(S) = S, gives a similarity solution of eq.(2.2) via 1 S = (C1 BesselJ(1,h) +C2 BesselH(1,h)). h Solving (3.12) for k(S) = S3 , gives a similarity solution of eq.(2.2) via √ √ C1 1 2 2 S = JacobiSN(( 1 − C1 lnh). i +C2 ,± − 1). h 2 2 −C12

(3.14)

(3.15)

III) For v3 = −z ∂∂y + y ∂∂z :

dy By integrating of the characteristic equation dt0 = dx0 = −z = dz = du0 , the corresponding invariants √ y follow as: t,x,y2 + z2 ,u. Substituting t = w, x = Y, z = Z − y2 , u(t,x,y,z) = s(w,Y,Z), reduces eq.(2.2) to

sYY − w2 sww + 4Ze−2Y sZZ − 3wsw + 4e−2Y sZ + 2sY − w2 k(s) = 0, The equation (3.16), has one infinitesimal symmetry ω1 = −2Z ∂∂Z + ∂∂Y .

15

(3.16)


• For ω1 : the corresponding invariants follow as: w, 12 lnZ +Y, s. Substituting w = f , Y = g− 21 lnZ, s(w,Y,Z) = T ( f ,g), reduces eq.(3.16) to − f 2 T f f + (1 + e−2g )Tgg − 3 f T f + 2Tg − f 2 k(T ) = 0.

(3.17)

The equation (3.17), do not has any infinitesimal symmetry. IV) For v4 = ∂∂y :

By integrating of the characteristic equation dt0 = dx0 = dy1 = dz0 = du0 , the corresponding invariants follow as: t,x,z,u. Substituting t = w, x = Y, z = Z, u(t,x,y,z) = s(w,Y,Z), reduces eq.(2.2) to (3.9).

4 Conclusion In this research, about the Gordon equation (2.2) related to the Milne metric, at first, its Lie reductions have been generally achieved. Then, its similarity solutions; for k(u) = un ;n = 0,1,3 have been acquired.

References [1] S.JAMAL , A.H.K ARA AND A.H.B OKHARI, Symmetries conservation laws and reduction of wave and Gordon-type equations on Riemannian manifolds, World Academy of Science, Engineering and Technology. 60 (2011). [2] www. wikipedia. org. [3] www. astronomy. swin. edu. au/ cosmos/ c/ Comoving Distance. [4] A. M ITRA, Friedmann-Robertson-Walker metric in curvature coordinatesand its applications, Grav. Cosmol. No 2, (2013). [5] G. B. S HAW,Lecture 7: Cosmic Distances, PHYS 652: Astrophysics. [6] P. P ETERSEN, Riemannian Geometry, springer, the United States of America, (2006). [7] H. D INGLE, On E. A.Milne’s theory of world structure and the expansion of universe, Imperial college of science and technology, (1988). [8] A.A.B YTSENKO , M.E.X.G UIMAR AES, R.K ERNER, Orbifold compactification and solutions of M-theory from Milne spaces, arXiv:hep-th/0501008v1 3 Jan, (2005). [9] P.J.O LVER, Applications of Lie groups to differential equations, Springer, NewYork, (1986).

16

M. Deldar Froutagheh :: Free paratopoligical groups

Paper No. 1.4

FREE PARATOPOLIGICAL GORUPS 1

Maryam Deldar Froutagheh e-mail: [email protected], Iran University of Zahedan, Sistan and Baluchestan, Iran. Abstract In this paper, we study the free paratopological groups FP(X) and AP(X) on a topological space X in the sense of Markov. We prove that FP(X) and AP(X) on an Alexandroff space X are Alexandroff spaces, we introduce description of neighborhood bases at identity for their topologies when the space X is Alexandroff and we give some properties on these neighborhood bases. Keywords: Alexandroff space, Free paratopological groups.

17


1 Introduction The free paratopological groups FP(X) and AP(X) on a topological space X in the sense of Markov are the free abstract groups Fa (X) and Aa (X) on X with the strongest paratopological group topologies on Fa (X) and Aa (X), respectively, that induce the original topology on X. This paper is adapted from ([1], chapter 3) and it is a study of the topology of free paratopological groups FP(X) and AP(X) on a space X. In this paper, we prove that the free paratopological groups FP(X) and AP(X) are Alexandro spaces if and only if the space X is Alexandro and then we introduce simple neighborhood bases at the identities of FP(X) and AP(X) for their topologies when the space X is Alexandro. We study some properties of these neighborhood bases and then as applications of this.

2 DEFINITIONS AND PRELIMINARIES Proposition 2.1. Let G be a group and let N be a collection of subsets of G, where each member of N cantains the identity element e of G. Then the collection N is a base at e for a paratopological group topology on G if and only if the following conditions are satisfied: 1. for all U,V ∈ N , there exists W ∈ N such that W ⊆ U ∩V ; 2. for each U ∈ N , there exists V ∈ N such that V 2 ⊆ U; 3. for each U ∈ N and for each x ∈ U, there exists V ∈ N such that xV ⊆ U and V x ⊆ U; and 4. for each U ∈ N and each x ∈ G, there exists V ∈ N such that xV x−1 ⊆ U. Definition 2.2. Let X be asubspace of a paratopological group G. Suppose that 1. the set X generates G algebraically, that is, hXi = G and 2. every continuouse mapping f : X −→ H of X to an arbitrary paratopological group H extends to a ontinuous homomorphism fˆ : G −→ H . Then G is called the Markov free paratopological group on X, and is denoted by FP(X). By substituting ”abelian paratopological group” for each occurrence of ”paratopological group” above we obtain the definition of the Markov free abelian paratopological group on X and we denote it by AP(X). Remark 2.3. We denote the free topology of FP(X) by τFP and the free topology of AP(X) by τAP and we note that the topologies τFP and τAP are strongest paratopological group topologies on the underlying sets of FP(X) and AP(X), respectively, that induce the original topology on X.

18


Definition 2.4. Let X be a topological space and α be an infinite cardinal. Following [4], we say that X T is a Pα -space if ϕ is open in X for each family ϕ of open subsets of X with |ϕ| ≤ α . Let α be an infinite singular cardinal and let X be a topological space. Then X is a Pα + -space if X is a Pα -space, where α + is the successor cardinal of α. Definition 2.5. A topological space X is said to be Alexandroff if the intersection of every family of open subsets of X is open in X. A space X is Alexandroff if and only if X is a Pα -space for every infinite cardinal α. Definition 2.6. Let G be a group and let H be a subset of G. Then we say that H is a submonoid of G if H contains the identity of G and closed under the multiplication in G. if, in addition, H satisfies ghg−1 ∈ H for all h ∈ H and g ∈ G, then we say that H is a normal submonoid of G. Theorem 2.7. Let (X,τ) be a topological space and let α + be the infinite successor cardinal of α. Then the collection of all sets which are the intersection of fewer than β open subset of X is a base for the topology τα on X, where β = α if α is regular and β = α + if α is singular. Proof. Let τ = {Ui }i∈I . We show that the collection β = { d∈D Ud : D ⊆ I and |D| < β } of subsets of X is a base for the topology τα on X, where β as defined in the statement of the theorem. It is well known that every infinite successor cardinal is regular, so in both cases, β is regular. We show that β is a base for some topology τ ∗ on X. If x ∈ X, there exists i0 ∈ I where x ∈ Ui0 and such T T that Ui0 ∈ β . Let B1 ,B2 ∈ β and let x ∈ B1 ∩ B2 . Assume that B1 = d∈D Ud and B2 = t∈T Ut , where T D,T ⊆ I and |D|,|T | < β . Let R = D ∪ T . So |R| < β . Hence B3 = r∈R Ur ∈ β and x ∈ B3 ⊆ B1 ∩ B2 . Therefore β is a base for some topology τ ∗ on X. We show second that (X,τ ∗ ) is a Pα -space. Let τ ∗ = {V j } j∈J and let M ⊆ J where |M| < β . Then we have T

\ m∈M

Vm =

\ [

[

=

(

m∈M Im , f (m)∈Im ∀m∈M

S

f :M→

Bm,i

m∈M i∈Im

\

Bm, f (m) ) ∈ τ ∗ ,

m∈M

where Im is an index set and Bm,i ∈ β for all m ∈ M and i ∈ Im . Thus τ ∗ contains τ and (X,τ ∗ ) is a Pβ -space, which implies that in both cases af β ,(X,τ ∗ ) is a Pα -space. ˆ is a Pα -space. Then in the case where α is Now let τˆ be a topology on X containing τ such that (X, τ) ˆ is a regular, we have β ⊆ τˆ and in the case where α is singular, by the argument above, we have (X, τ) ˆ Thus τ ∗ ⊆ τˆ and hence τ ∗ is the smallest topology on X containing Pα + -space, which implies that β ⊆ τ. τ such that (X,τ ∗ ) is a Pα -space. Therefore, τ ∗ = τα .

19


Proposition 2.8. Let (G,τ) be a paratopological group . Then (G,τα ) is a paratopological group. Proof. Let g1 ,g2 ∈ G and let U ∈ τα contain g1 g2 . We show that there exist U1 ,U2 ∈ τα containing g1 ,g2 , respectively, such that U1U2 ⊆ U. Now by Theorm 2.7 there is a set Λ, where |Λ| < β and β is as in the theorem such that g1 g2 ∈ Uλ for all λ ∈ Λ. Thus g1 g2 ∈ Uλ for all λ ∈ Λ. Since τ is a paratopological group topology on G, for each λ ∈ Λ, there are V (λ ),W (λ ) ∈ τ containing g1 ,g2 , repectively, T T such that V (λ )W (λ ) ⊆ Uλ . Let U1 = λ ∈Λ V (λ ) and U2 = λ ∈Λ W (λ ). Then U1U2 ⊆ Uλ for all λ ∈ Λ. T Hence, U1 ,U2 ∈ τα and U1U2 ⊆ λ ∈Λ Uλ ⊆ U. Therefore, τα is a paratopological group topology on G. Proposition 2.9. Let (Y,τY ) be a subspace of a topological space (X,τ). Then τα|Y = (τY )α . Theorem 2.10. Let X be a topological space. Then the free paratopological group FP(X) on X is a Pα -space if and only if the space X is a Pα -space. Proof. =⇒ : Assume that FP(X) is a Pα -space. Since X is a subspace of FP(X), it is easy to see that X is a Pα -space. ⇐= : Let τ be the topology of X and let τFP be the free topology of FP(X). We show that (τFP )α = τFP . By Proposition 2.8, (τFP )α is a paratopological group topology on Fa (X) and it is stronger than τFP . However, τFP is the free paratopological group topology on Fa (X), which is the strongest paratopological group topology on Fa (X) inducing the original topology τ on X. By Proposition 2.9, we have (τFP )α |X = (τFP |X)α and since (τFP |X)α = (τ)α = τ, (τFP )α induces the topology τ of X. Thus we have (τFP )α = τFP and therefore, FP(X) is a Pα -space. The same result oF Theorem 2.10 is true for AP(X). Corollary 2.11. The free paratopological group FP(X) (AP(X)) on a space X is an Alexandroff space if and only if X is an Alexandroff space. Proposition 2.12. If H is a normal submonoid of a group G, then {H} is a neighborhood base at the identity of G for a paratopological group topology on G. Proof. Let H be a normal submonoid of G. Then it is easy to see that {H} satisfies the conditions of base for paratopological group and then {H} is a neighborhood base at the identity of G for a paratopological group topology on G. Proposition 2.13. Let X be an Alexandroff space. Then the neighborhood base at the identity e (0A ) in FP(X) (AP(X)) is a single normal submonoid. Proof. By corollary 2.11, the group FP(X) is an Alexandroff space. Let U(e) be the minimal open neighborhood of the identity element e in FP(X). so {U(e)} is a local base at e for the free topology of FP(X). Now there exists a neighborhood V of e in FP(X) such that V 2 ⊆ U(e). since U(e) ⊆ V ,

20


U(e)2 ⊆ V 2 ⊆ U(e). Therefore, U(e) is a submonoid. Now if g ∈ FP(X), there exists a neighborhood W of e such that gW g−1 ⊆ U(e). since U(e) is the minimal open neighborhood of e, then U(e) ⊆ W , which implies that gU(e)g−1 ⊆ gW g−1 ⊆ U(e). Therefore, U(e) is a normal submonoid of FP(X). Similarly, we can prove the statement of the theorem for AP(X). Proposition 2.14. Let X be an Alexandroff space. Then the topologies OF and OA induce topologies coarser than the original topology on X. Proof. We prove that OF|X induces a topology coarser than the original topology on X. Let x ∈ X and let y ∈ U(x). Then x−1 y ∈ NF . This implies that y ∈ xNF ∩ X and then U(X) ⊆ xNF ∩ X. Thus OF|X is coarser than the original topology on X. an analogous proof can be used for OA . Theorem 2.15. Let X be an Alexandroff space. Then NF (NA ) is a neighborhood base at e (0A ) for the topology of FP(X)(AP(X)). Proof. We prove the theorem for NF , since the proof for NA is the same. We show first that the topology OF is finer than the free topology τFP of FP(X). Let ξ : X → G be a continuous mapping of the space X into an arbitrary paratopological group G. Then ξ extends to a homomorphism ξˆ : Fa (X) → G. We show that ξˆ is continuous with respect to the topology OF . Let V be a neighborhood of ξˆ (e) = eG in G.Fix x ∈ X. Then ξ (x)V is a neighborhood of ξ (x) in G. Since ξ is continuous at x, ξ (U(x)) ⊆ ξ (x)V and Since ξˆ |X = ξ , ξˆ (U(x)) ⊆ ξˆ (x)V . Because ξˆ is a homomorphism, ξˆ (x−1U(x)) ⊆ V . Since x is any point in X, we have ξˆ (

[

x−1U(x)) ⊆ V.

(2.1)

x∈X

Fix n ∈ N. Then there exists a neighborhood u of eG in G such that U n ⊆ V and also, for all g ∈ Fa (X), there exists a neighborhood W of eG in G such that ξˆ (g)W (ξˆ (g))−1 ⊆ W . Since V is any neighborhood of eG in G, from (2.1), we have ξˆ (bigcupx∈X x−1U(x)) ⊆ W . Fix g ∈ Fa (X). So we have ξˆ (g)ξˆ (

[

x−1U(x))(ξˆ (g))−1 ⊆ ξˆ (g)W ξˆ (g)−1 .

x∈X

Since ξˆ is a homomorphism, ξˆ (

[

gx−1U(x)g−1 ) ⊆ ξˆ (g)W ξˆ (g)−1 ⊆ U.

x∈X

Since (2.2) holds for every g ∈ Fa (X), we have ξˆ (

[

[

gx−1U(x)g−1 ) ⊆ U.

g∈Fa (X) x∈X

21

(2.2)


Thus we have ξˆ ((

[

[

gx−1U(x)g−1 )n ) ⊆ U n ⊆ V.

g∈Fa (X) x∈X

Since n is any element of N, ξˆ (

[

(

[

[

gx−1U(x)g−1 )n ) ⊆ V.

n∈N g∈Fa (X) x∈X

S S S Since NF = n∈N ( g∈Fa (X) x∈X gx−1U(x)g−1 )n , we have ξˆ (NF ) ⊆ V . Thus ξˆ is continuous with respect to the topology OF and therefore, OF is finer than τFP . By proposition 2.14 OF|X is coarser than the original topology on X. Since OF is finer that τFP ,OF|X induces the original topology on X. Thus we satisfied the conditions of Definition 2.2, which implies that OF = τFP . Therefore, NF is a neighborhood base at e of the group FP(X).

References [1] A LI S AYED E LFARD, Free paratopological groups, PhD Thesis, University of Wollongong, Australia (2012). [2] O. V. S IPACHEVA and M. G. T KACENKO, Thin and bounded subsets of free topological groups, Seminar on General Topology and Topological Algebra (Moscow, 1988/1989), Topology Appl., Topology and its Applications,vol. 36, no. 2, (1990), pp.165-176. [3] R. KOPPERMAN, Asymmetry and duality in topology, Topology Appl., vol. 66 (1995), pp. 1-39. [4] F. G. A RENAS, Alesandroff spaces, Acta Math. Univ. Comenian. (N.S.), vol. 68 (1999), pp. 1725.

22

M. Parhizkar and P. Bahmandoust and D. Latifi :: Biinvariant Finsler metrics on Lie groups

Paper No. 1.5

Bi-invariant Finsler Metrics on Lie Groups Mojtaba Parhizkar1 and Parisa Bahmandoust2 and Dariush Latifi3 e-mail: [email protected], Iran University of Mohaghegh Ardabili, Ardabil, Iran. 2 e-mail: [email protected], Iran University of Mohaghegh Ardabili, Ardabil, Iran. 3 e-mail: [email protected], Iran University of Mohaghegh Ardabili, Ardabil, Iran. 1

Abstract In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics. We first show that every compact Lie group admits a bi-invariant Finsler metric. Finally we give an explicit formula for the flag curvature of bi-invariant Finsler metrics Keywords: Invariant Finsler metric, Bi-invariant metric, Lie group, Flag curvature, geodesic

1 Introduction The geometry of invariant Finsler structures on homogeneous manifolds is one of the interesting subjects in Finsler geometry which has been studied by some Finsler geometers, during recent years. Lie groups are the most beautiful and most important manifolds. On the one hand, these spaces contain many prominent examples which are of great importance for various branches of mathematics, like homogeneous spaces, symmetric spaces, and Grassmannians. Lie groups are, in a sense, the nicest examples of manifolds and are good spaces on which to test conjectures (Milnor, J., 1976). Therefore it is important to study invariant Finsler metrics. S. Deng and Z. Hou studied invariant Finsler metrics on homogeneous spaces and gave some descriptions of these metrics (Deng, S. and Z. Hou, 2004; Deng, S. and Z. Hou, 2004). There is a recent paper on invariant Finsler metrics on two-step nilpotent Lie groups (Toth, A. and Z. Kovacs, 2008). Also, in (Latifi, D. and A. Razavi, 2006; Latifi, D., 2007; Latifi, D. and A. Razavi, 2009) we have studied the homogeneous Finsler spaces and the homogeneous geodesics in homogeneous Finsler spaces.(for example see [2], [3], [4], [5].). Among the invariant metrics the bi-invariant ones are the simplest kind. They have nice and simple geometric properties, but still form a large enough class to be of interest. In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics.We first show that every compact Lie group admits a non-Riemannian bi-invariant Finsler metric. Definition 1.1. Finsler metric on M is a non-negative function F : T M −→ R which has the following properties: (1) F is smooth on the slit tangent bundle T M\{0}, (2) F(x,λ y) = λ F(x,y) for all x ∈ M, y ∈ T M and λ > 0,

23


2 2

(3) the n × n Hessian matrix [gi j (x,y)] = [ 21 ∂∂yi ∂Fy j ] is positive definite at every point (x,y) ∈ T M . The G-invariant Finsler functions on T G may be identified with the Minkowski norms on g. If F˜ : T G −→ ˜ R+ is an G-invariant Finsler function, then, we may define F : g −→ R+ by F(X) = F(e,X), where e ˜ denotes the identity in G. Conversely, if we are given a Minkowski norm F : T G −→ R+ , then F˜ arises ˜ from an G-invariant Finsler function F : T G −→ R+ given by F(g,X) = F(X) for all (g,X) ∈ G × g.

2 Bi-invariant Finsler metrics on Lie groups Definition 2.1. A Finsler metric on G that is both leftinvariant and right-invariant is called bi-invariant. Let G be a compact Lie group. Fix a base ω1 ,ω2 ,...,ωn in Te M and put ω = ω1 ∧ ω2 ∧ ... ∧ ωn where n = dimG. Extend ω to a left-invariant differential form Ω on G byputting Ωg = (Lg)∗ ω . The form Ω never vanishes. The form determines an orientation of G. Recall that a chart with coordinates x1 ,...,xn is called positively if dx1 ∧ dx2 ∧ ... ∧ dxn = f Ω where f is a positive function defined on the coordinate neighborhood. Clearly, the atlas consisting of all positively oriented charts determines an orientation on G. Indeed, if dy1 ∧ dy2 ∧ ... ∧ dyn = hΩ and h > 0, then: h dy1 ∧ dy2 ∧ ... ∧ dyn = dx1 ∧ dx2 ∧ ... ∧ dxn f

(2.1)

On the other hand, dy1 ∧ dy2 ∧ ... ∧ dyn =

∂ (y1 ,...,yn ) 1 dx ∧ dx2 ∧ ... ∧ dxn ∂ (x1 ,...,xn )

(2.2)

So ∂ (y1 ,...,yn ) h = >0 ∂ (x1 ,...,xn ) f

(2.3)

Now, for any a ∈ G, we can easily see that R∗a Ω is left invariant. It follows that R∗a Ω = f (a)Ω. We can easily see that f (ab) = f (a) f (b) that is f : G −→ R\{0} is a continuous homomorphism of G into themultiplicative group of real numbers. Since f (G) is compact connected subgroup, the conclusion f (G) = 1 holds. Therefore, R∗a Ω = Ω. So Ω is bi-invariant volume element on G. Theorem 2.2. Every compact Lie group admits a biinvariant Finsler metric. Proof. Let Ω be the bi-invariant volume element. Let Fe be a Minkowski norm on Te G = g. Then, define the functionF˜e on Te G by: F˜e2 (X) =

Z

G

Fe2 (AdgX)Ω

24

(2.4)


Let K1 ,...,KN be a covering of G by cubes, and that let ϕ1 ,...,ϕN be a corresponding partition of unity, and Ω = Ω123...n dx1 ∧ ... ∧ dxn . Now we can write: F˜e2 (X) =

Z

G

Fe2 (AdgX)Ω =

N

∑ n! 1

Z

Ki

Fe2 (ϕi Adg X)Ω12...n dx1 ∧ ... ∧ dxn

(2.5)

By definition of the orientation, Ω12...n > 0, and ϕi (x) positive in K ; furthermore, Fe2 (Adg X)Ω > 0 R 2 i if X 6= 0. Since all summands in the above expression for G Fe (Adg X)Ω are positive, we come to R 2 ˜ ˜ ˜ G Fe (Adg X)Ω > 0. So, Fe is welldefined function F : Te G −→ [0,∞), Fe (X) > 0 if X 6= 0. We can easily ˜ ˜ ˜ see that Fe (X) = 0 if and only if X = 0. Clearly Fe (λ X) = λ Fe (X) for any X ∈ Te G, λ > 0. Since Fe is C∞ on Te G\{0}, we see that F˜e is C∞ on Te G\{0}. Now for any y 6= 0, u, v ∈ Te G by a direct computation, we have: g˜y (u,v) =

Z

gAdg (y) (Adg (u),Adg (v))Ω.

(2.6)

˜ is a Minkowski norm on Te G. In Since gAdg (y) is positively definite, hence g˜y is positive definite. So F(e) the following, we show that F˜e is Ad(G)invariant. F˜e2 (Adh X) =

Z

G

Fe2 (Adg Adh X)Ω =

Z

G

Fe2 (Adgh X)Ω.

(2.7)

The diffeomorphism Lh and Rh preserve the orientation because Ω is bi-invariant. So, for the diffeomorphism Ih = Rx−1 ◦ Lx : G −→ G, we have: Z

Ih (G)

Fe2 (Adg X)Ω =

Z

G

Fe2 (Adgh X)R∗h Ω.

(2.8)

Since Ih (G) = G and R∗h Ω = Ω, we see that: F˜e2 (Adh X) = F˜e2 (X).

(2.9)

So F˜e (Adh X) = F˜e (X). ˜ Extend the Minkowski norm in Te G, thus defined to a left-invariant Finsler metric on G, Thus, F(X) = ˜Fe ((La−1 )∗a X), for all X ∈ Ta G. We show that this Finslermetric is bi-invariant.We only need to check the right-invariance, we have: ˜ a )∗ X) = F((L ˜ a−1 )∗ (Ra )∗ X). F((R

(2.10)

˜ Consequently, by Ad-invariance of F: ˜ a )∗ X) = F(Ad ˜ a−1 X) = F(X). ˜ F((R

25

(2.11)


In [3], D. Latifi has proved follows: Theorem 2.3. Let G be a connected Lie group with Lie algebra g, and let F be a left-invariant Finsler metric on G. Then X ∈ g − {0} is a geodesic vector if and only if: gX (X,[X,Y ]) = 0,

∀Z ∈ g.

Lemma 2.4. Let F be a bi-invariant Finsler metric on a connected Lie group. Then for every 0 6= 0,z ∈ g we have: gy (y,[y,z]) = 0 Proof. See [4]. Since F is bi-invariant, we have: gy ([x,u],v) + gy (u,[x,v]) + 2Cy ([x,y],u,v) = 0 It follows from the homogeneity of F that Cy (y,u,v) = 0, for every 0 6= y, u, v, ∈ g. So we have gy (y,[y,x]) = 0.

Corollary 2.5. If G is a Lie group endowed with bi-invariant Finsler metric, then, the geodesics through the identity of G are exactly oneparameter subgroups. Corollary 2.6. Every compact Lie group admits a Finsler metric, then, the geodesics through the identity of G are exactly one parameter subgroups. In [4], D. Latifi has proved follows: Theorem 2.7. Let G be a Lie group with a bi-invariant Finsler metric F . Then the Finsler metric F is of Berwald type. Corollary 2.8. Every compact Lie group admits a Finsler metric, then, the Finsler metric is of Berwald type. Corollary 2.9. Every compact Lie group admits a Finsler metric F, then we have: ∀x,y ∈ g. 1) ∇x y = 21 [x,y], 1 ∀x,y,z ∈ g. 2)R(x,y)z = [[x,y],z], 4 3)Then the flag curvature of the flag (P,y) such that {u,y} is a basis of P in g given by K(P,y) =

gy ([u,y],[u,y]) 1 4 gy (y,y)gy (u,u) − g2y (u,y)

3 Conclusion In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics. Finally we give an explicit formula for the flag curvature of bi-invariant Finsler metrics.

26


References [1] J. MILNOR, Curvature of left invariant metrics on Lie groups, Advan. math, Vol. 21 (1976), 293392. [2] S. D ENG Z. H OU, Invariant Randers Metrics on Homogeneous Riemannian Manifolds, J. Phys. A: Math. Gen., Vol. 37 (2004), 4353-4360. [3] D. L ATIFI, Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phys., Vol. 57 (2007), 1421-1433. [4] D. L ATIFI A. R AZAVI, Bi-invariant Finsler Metrics on Lie Groups, Australian Journal of Basic and Applied Sciences, Vol. 5 (2011), 507-511. [5] D. L ATIFI A. R AZAVI, On homogeneous Finsler spaces, Rep. Math. Phys., Vol. 57 (2006), 357366. [6] D. BAO SS. CHERN Z. NEW-YORK (2000).

SHEN,

An introduction to Riemann-Finsler geometry., Springer-Verlag,

27

E. Keyhani :: Generalized differentiation in modules over topological ∗-algebras

Paper No. 1.6

GENERALIZED DIFFERENTIATION IN MODULES OVER TOPOLOGICAL ∗-ALGEBRAS E. KEYHANI Abstract. In this paper, at first we give a definition of a generalized total differentiation and A-linear generalized differentiation and skew-linear generalized differentiation and then we extended some results on differentiation in modules over topological ∗-algebras to this new extended theory.

1. Introduction In recent years, there has been a systematic development of the theory of topological ∗algebras. This extension is justified by its applications in pure mathematics. Specially a C ∗ -algebra were considered by T. K. Kandelaki, K. Fujii, and A. S. Miscenko and YU .P .Solovev. Note also that certain results are still valid for more general topological algebras or even topological rings. We summarize here the basic concepts and results concerning the topologies of the spaces used in this paper. By a topological ring R we mean a ring R provided with a topology making addition and multiplition jointly continuous. Similarly by a topological R-module M we mean an R-module M equipped with a topology making addition and scalar multiplition jointly continuous. Let R be a topological ring with unit. We denote by P(R) and M od(R) the categories of projective finitely generated R-module and topological R-modules, respectively. If M ∈ P(R), then by definition there exists M ′ ∈ P(R) and m ∈ N with M ⊕ M ′ ∼ = Rm , where ∼ = denotes an isomorphism of R-modules. Consider Rm provided with the product topology and M with the relative topology denoted by τM . It is proved that τM does not depend either on M ′ or m. We call τM the canonical topology of M . An equivalent definition of the canonical topology is given in [?]. One can check that all the aforementioned topologies coincide on projective finitely generated modules over an algebra as above. We 2010 Mathematics Subject Classification. Primary 16W25, Secondary 46KXX,46H25, 46HXX . Key words and phrases. Total σ-differentiation, A-linear σ-differentiation, Skew linear σ-differentiation.

28


have the following.

Theorem 1.1. Let M ∈ P(R). Then (i) τM is the strongest topology making M a topological R-module. (ii) τM is the weakest topology making every R-linear map f : M → N continuous, for any N ∈ M od(R).

Therefore, τM is the unique topology having both the above properties. If A is a ∗-algebra and M, N are A-modules, a map f : M → N is said be skew-linear,if f is additive and f (ax) = a∗ f (x), for any a ∈ A and x ∈ M . For a topological ∗-algebra ( a topological algebra with a continuous involution), the continuity of A-linear maps is extended to that of multi-linear maps with partial maps either A-linear or skew-linear. Proposition 1.2. Let A be a commutative topological ∗-algebra with unit, M1 , · · · , Mk ∈ P(A), N ∈ M od(A) and let f : M1 × · · · MK → N be a map. If, in each variable, f is either A-linear or skew-linear, then f is continuous. Proof. Without loss of generality, we assume that f is A-linear with respect to each of the first p variables and skew- linear with respect to each of the last q := k − p ones. (i) Suppose first that each Mi is a free finitely generated A-module, that is, Mi ≃ Aν(i) , ν(i) ∈ N, i = 1, . . . , k. Let {ej : j = 1, . . . , ν(1) + . . . + ν(k)} be the canonical basis of M1 ×. . .×Mk . Then ∑ ∑ ∑ f ( j xj ej ) = a xa f (ea )+ b x∗b f (eb ), where 1 ≤ a ≤ ν(1)+. . .+ν(p) < b ≤ ν(1)+. . .+ν(k). Obviously, f is continuous. (ii) If now Mi ∈ P(A) , there are Mi′ ∈ P(A) and ν(i) ∈ N, with Mi ⊕ Mi′ ≃ Aν(i) . Therefore, f is extended to an f : A∼(1) × . . . × A∼(k) → N, having partial maps either A-linear or skew-linear. By (i), f is continuous, hence, so is f with respect to τM .

□

2. Main Theorem In the sequel, A is a topological ∗-algebra with unit. For any M, N ∈ M od(A) and x ∈ M , we denote by 0M the zero element of M , by N (x) the set of open neighbourhoods of x, and by LA (M, N ) the set of A-linear maps f : M → N

29


Definition 2.1. Let M, N ∈ M od(A), Ω ∈ N (0M ), and ϕ : Ω → N and σ : A → A are maps we say that ϕ is σ-infinitesimal, if ∀V ∈ N (0N ) ∃U ∈ N (0M ) ∗

∀B ∈ N (0A ) ∃A ∈ N (0A ) ∗

ϕ(aU + a U ) ⊂ σ(a)BV + σ(a )BV,

∀a ∈ A

(1)

The set of the above maps will be denoted by Rσ (M, N ). In the sequel, every projective finitely generated module over a unital topological ring will be provided with the canonical topology. Assume now that the ring of coefficients is a unital locally m-convex (abbreviated lmc) ∗algebra A and let ΓA denote a calibration of A. The product topology of AM is induce by the family P ∼ : AM → R : (a1 , . . . , am ) →

∑

p(ai );

p ∈ ΓA .

i

If M ∈ P(A) is a direct summand of Am , then τM is defined by the family of restrictions p∼ |M ;

p ∈ ΓA

i.e., the topological A-module (M, τM ) is, in addition, a locally convex space. To simplify notations, we write p∼ instead of p∼ |M . Since every p ∈ ΓA is submultiplicative, one has that p∼ (ax) ≤ p(a)p∼ (x); x ∈ M,

a ∈ A.

(2)

Definition 2.2. Let M, N ∈ M od(A), x ∈ M, Ω ∈ N (x), and f : Ω → N a map. We say that f is totally σ-differentiable at x, if there exist a continuous A-linear map Df (x) : M → N which is called A-linear σ-differential of f at x and a continuous skew-linear map Sf (x) : M → N which is called skew σ-differential of f at x such that the remainder of f at x, i.e., the map ϕ(h) := f (x + h) − f (x) − Df (x)(h) − Sf (x)(h)

(3)

is σ-infinitesimal. We call T f (x) := Df (x) + Sf (x) the total σ-differential of f at x. If Sf (x) = 0, f is said to be A-linearly σ-differentiable at x and if Df (x) = 0, f is said to be A-linearly skew σ-differentiable at x. we denote by Tσ Dx (M, N ), Lσ Dx (M, N ) and Sσ Dx (M, N ) the setes of N -valued totally, Alinearly and skew σ-differentiable maps at x, respectively.

30


Proposition 2.3. Let A be a topological ∗-algebra with unit and M, N ∈ M od(A) and σ : A → A be an additive and multiplicative mapping.If x ∈ M and f ∈ Tσ Dx (M, N ), then f is continuous at x. Proof. It suffices to prove that every ϕ ∈ Rσ (M, N ) is continuous at 0M with ϕ(0M ) = 0N . Indeed, if V ∈ N (0N ), continuity of module operations implies the existence of a V1 ∈ N (0N ) and a balanced B1 ∈ N (0A ) with B1 V1 + B1 V1 ⊆ V . Since ϕ ∈ Rσ (M, N ), there exist U1 ∈ N (0M ) and A ∈ N (0A ) such that ϕ(au1 + a∗ u1 ) ⊂ σ(a)B1 V1 + σ(a∗ )B1 V1 , ∀a ∈ A. On the other hand, there is an ϵ ∈ A ∩ {0, 1}. Thus, for U := ϵU1 + ϵU1 ∈ N (0M ), we have ϕ(U ) ⊂ V . In fact ϕ(ϵU1 + ϵU1 ) ⊂ σ(ϵ)B1 V1 + σ(ϵ∗ )B1 V1 = B1 V1 + B1 V1 ⊆ V . Which proves □

the assertion.

Lemma 2.4. Let A be a lmc ∗-algebra with unit and N ∈ P(A). Then, for every V ∈ N (0N ), there exists V1 ∈ N (0N ) such that, for every B ∈ N (0A ), there exists B1 ∈ N (0A ) with B1 V1 + B1 V1 ⊂ BV

(4).

Proof. Let V ∈ N (0N ). There exist a seminorm P ∼ on N , induced by a P ∈ ΓA , and ϵ > 0 with SP ∼ (0N , ϵ) ⊂ V . We set B1 := SP (0A , 2δ ). Using (2), one checks that (4) is satisfied. □ Lemma 2.5. (i) Let A be a topological ∗-algebra with unit, M, N ∈ M od(A), ϕ ∈ Rσ (M, N ), and a ∈ Z(A) (centre of A). Then aϕ ∈ Rσ (M, N ). (ii) Let A be a lmc ∗-algebra with unit, M ∈ M od(A), N ∈ P(A), and ϕ, ψ ∈ Rσ (M, N ). Then ϕ + ψ ∈ Rσ (M, N ). Proof. (i) This is straightforward. (ii) Let V ∈ N (0N ) and let V1 be the corresponding neighbourhood of 0N in the preceding lemma. Since ϕ, ψ ∈ Rσ (M, N ), there exist Uϕ , Uψ ∈ N (0M ) satisfying (1) with respect to V1 . We set U := Uϕ ∩ Uψ . Let now B ∈ N (0A ). Besides, the B1 ∈ N (0A )) with B1 V1 + B1 V1 ⊂ BV . Then, there are Aϕ , Aψ ∈ N (0A ) with ϕ(aUϕ + a ∗ Uϕ ) ⊂ σ(a)B1 V1 + σ(a∗)B1 V1 , ∀a ∈ Aϕ , ψ(aUψ + a ∗ Uψ ) ⊂ σ(a)B1 V1 + σ(a∗)B1 V1 , ∀a ∈ Aψ . If A := Aϕ ∩ Aψ , then we check that (ϕ + ψ)(aU + a ∗ U ) ⊂ σ(a)BV + σ(a∗)BV

31


holds, for every a ∈ A.

□

Proposition 2.6. (i) Let A be a topological ∗-algebra with unit, M, N ∈ M od(A), and x ∈ M . If f ∈ Tσ Dx (M, N ) and a ∈ Z(A), then af ∈ Tσ Dx (M, N ) and D(af )(x) = aDf (x), S(af )(x) = aSf (x). (ii) Let A be a lmc ∗-algebra with unit, M ∈ M od(A), N ∈ P(A), and x ∈ M . If f, g ∈ Tσ Dx (M, N ), then f + g ∈ Tσ Dx (M, N ) and D(f + g)(x) = Df (x) + Dg(x), S(f + g)(x) = Sf (x) + Sg(x). Corollary 2.7. Let A be a commutative lmc ∗-algebra with unit, M ∈ M od(A), and N ∈ P(A). Then (i) Rσ (M, N ) is an A-module. (ii) Tσ Dx (M, N ), Lσ Dx (M, N ) and Sσ Dx (M, N ) are A-modules, for every x ∈ M . References [1] A.Mallios, Hermitian k-theory over topological ∗-algebras, J.Math. Anal. 106(1985),454-539. [2] A.Mallios, Continuous vector bundles over topological algebras, J.Math.Anal.Appl.132(1986),245254. [3] A.Mallios, Topological Algebras: Selected Topics , North-Holand, Amesterdam, 1986. [4] A.Mallios, Continuous vector bundles over topological algebras, J.Math.Anal.Appl.132(1988),401423. [5] M.H. Papatriantafillou, Differentiation in modules over topological ∗-algebras , Journal of mathematical analysis and application 170, 255-275(1992). [6] M.H. Papatriantafillou, Methods of differentiation in topological A-modules, Bull. Soc. Math. Grece 27(1986), 95-110. [7] M. H. Papatriantafillou, Translation invariant topologies on commutative ∗-algebras, Period Mad Hungar. 23 (3) (1991), 185-193. [8] V.L.Averbukh and O.G. Smolyanov ,, The various definitions of the derivative in linear topological spaces, Russian Math, Surveys 23, No. 4(1968),67-113. Academic Member of Department of Mathematics, Islamic Azad University, Hamedan Branch, Hamedan, Iran. E-mail address: [email protected].

32

A. Babaee, B. Mashayekhy and H. Mirebrahimi :: On Hawaiian groups of pointed spaces

Paper No. 1.7

ON HAWAIIAN GROUPS OF POINTED SPACES Ameneh Babaee1 and Behrooz Mashayekhy2 and Hanieh Mirebrahimi3 1 e-mail: [email protected], 2 e-mail: [email protected], 3 e-mail: h [email protected], 1,2,3 Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran. Abstract In this paper we investigate some behaviours of Hawaiian groups of pointed spaces with respect to their base points. It is known that Hawaiian groups are not independent to the choice of the points in general. We present some conditions of spaces whose Hawaiian groups are independent of the choice of base point. Keywords: Hawaiian Earring, n-dimensional Hawaiian Earring, Homotopy group, Hawaiian group.

1 Introduction and Motivation One-dimensional Hawaiian Earring H1 is defined to be the union of circles in the Euclidean plane R2 with center (1/n,0) and radius 1/n for n = 1,2,3,..., equipped with the subspace topology.

Figure 1: One-dimensional Hawaiian Earring.

33


In 2000, Hawaiian Earring extended to higher dimension. Eda et. al. [2] defined the n-dimensional Hawaiian Earring, n ∈ N, as the following subspace of the (n + 1)-Euclidean space R(n+1) Hn = {(r0 ,r1 ,...,rn ) ∈ R(n+1) | (r0 − 1/k)2 + ∑i=1 ri2 = (1/k)2 ,k ∈ N}, n

which is the union of n-dimensional spheres Skn , with center (1/k,0,...,0) ∈ Rn+1 and radius 1/k for k = 1,2,3,.... Here θ = (0,0,...,0) is regarded as the base point of Hn .

Figure 2: Two-dimensional Hawaiian Earring. In 2006, Karimov et. al. [3], using n-dimensional Hawaiian Earring, for n ∈ N, defined a new notion, the n-th Hawaiian group of a pointed space (X,x0 ) to be the set of all pointed homotopy classes [ f ], where f : (Hn , θ ) → (X,x0 ) is continuous, with a group operation which comes naturally from the operation of n-th homotopy group denoted by Hn (X,x0 ). This correspondence induces a covariant functor Hn : hTop∗ → Groups, from the pointed homotopy category, hTop∗ , to the category of all groups, Groups, for n ≥ 1. There exists a natural relation between the Hawaiian groups and the homotopy groups of a pointed space (X,x0 ). Karimov et. al. defines homomorphism φ : Hn (X,x0 ) → ∏k∈N πn (X,x0 ), with the rule φ ([ f ]) = ([ f |Sn ],[ f |Sn ],[ f |S3n ],...). The authors proved that the homomorphism φ can be a monomorphism in some 1 2 senses and they present an inverse for it [1]. In [3] also some advantages of Hawaiian group functor is presented rather than other famous functors such as homotopy, homology and cohomology functors. In particular, there exists a contractible space with non-trivial 1-Hawaiian group. Example 1.1 ([3]). Let C(H1 ) be the cone over H1 which is contactible space, hence all its homotopy, homology and cohomology groups are trivial, but it is showed that H1 (C(H1 ), θ ) is uncountable [3]. The authors [1, Theorem 2.13], presented the structure of Hawaiian groups of any cone as follows.

34


Theorem 1.2 ([1]). Let CX denote the cone over a space X, then Hn (X,x0 ) Hn (CX, x˜t ) ∼ , = w ∏i∈N πn (X,x0 ) when x˜t = [x0 ,t] and t ̸= 1. It is known that if t = 1, then x˜t is the vertex of the cone CX. So (CX, x˜1 ) is pointed homotopy equivalent to a point whose Hawaiian groups are trivial. Also, this functor can help us to get some local properties of spaces. In fact, if X has a countable local basis at x0 , then countability of the n-Hawaiian group Hn (X,x0 ) implies n-locally simply connectedness of X at x0 (see [3, Theorem 2]). By n-locally simply connectedness of X at x0 , we mean each open neighbourhood U of x0 contains an open neighbourhood V of x0 such that the homomorphism πn (V,x0 ) → πn (U,x0 ) induced by the inclusion is trivial. In the following Theorem, the authors give some equivalent conditions for n-locally simply connectedness. Theorem 1.3 ([1]). Let (X,x0 ) be a first countable pointed space and n ≥ 1, then the following statements are equivalent. (i) X is n-locally simply connected at x0 . (ii) φ : Hn (X,x0 ) → ∏w πn (X,x0 ) is an isomorphism. (iii) Hn (C(X), xˆt ) is trivial, where C(X) is the cone over X, t ̸= 1 and xˆt = (x0 ,t). By the above fact, this functor has useful advantages than homotopy group. For instance, it is known that in a path connected space, all homotopy groups are independent of the choice of points, but there exist some examples of path connected spaces with non-isomorphic Hawaiian group at several points (see Example 1.1). Consequently, if two points x1 ,x2 ∈ X satisfy Hn (X,x1 ) ∼ ̸ Hn (X,x2 ), then there is no pointed homotopy = equivalence between (X,x1 ) and (X,x2 ). In this paper, we establish some conditions under which the Hawaiian groups of (X,x1 ) and (X,x2 ) with distinct points x1 ,x2 ∈ X are isomorphic.

2 Main Results Definition 2.1. Let (X,x0 ) and (Y,y0 ) be two pointed topological spaces. We say that (X,x0 ) and (Y,y0 ) are semi-locally Hn -isomorphic, if there exist open neighbourhoods U of x0 and V of y0 such that Hn (U,x0 ) ∼ = Hn (V,y0 ). Remark 2.2. Let (X,x0 ) and (Y,y0 ) be two pointed spaces and let there exist neighbourhoods U of x0 , V of y0 and a pointed homotopy equivalence f : (U,x0 ) → (V,y0 ). Then f induces isomorphism Hn ( f ) : Hn (U,x0 ) → Hn (V,y0 ) and so (X,x0 ) and (Y,y0 ) are semi-locally Hn -isomorphic.

35


Remark 2.3. By [1], for n ≥ 2 and θ ′ ̸= θ , we have Hn (Hn , θ ) ∼ ̸ Hn (Hn , θ ′ ). = So there is no pointed homotopy equivalence from (Hn , θ ) to (Hn , θ ′ ), for θ ′ ̸= θ . Because we know that Hn : hTop∗ → Groups is a functor and hence if two spaces (X,x0 ) and (Y,y0 ) are pointed homotopy equivalent, then Hn (X,x0 ) ∼ = Hn (Y,y0 ). This fact has a more strong version for homotopy groups; if two spaces (X,x0 ) and (Y,y0 ) are freely homotopy equivalent, then πn (X,x0 ) ∼ = πn (Y,y0 ), but it is not authentic for Hawaiian groups. In the following theorem, we intend to have a similar result. Theorem 2.4. Let (X,x0 ) and (Y,y0 ) be two semi-locally Hn -isomorphic pointed spaces, and also let πn (X,x0 ) ∼ = πn (Y,y0 ), then Hn (X,x0 ) ∼ = Hn (Y,y0 ). Sketch of proof. Let U and V be neighbourhoods of x0 and y0 respectively, and let h : πn (X,x0 ) → πn (Y,y0 ) and f : Hn (U,x0 ) → Hn (V,y0 ) be isomorphisms. Now let [α ] ∈ Hn (X,x0 ), since α : (Hn , θ ) → (X,x0 ) is continuous, there exists K ∈ N such that if k ≥ K, then Im(α |Skn ) ⊆ U. Let K be the minimum integer for all elements of [α ]. Define ψ : Hn (X,x0 ) → Hn (Y,y0 ) by ψ ([α ]) = [β ], in which β |Skn = βk such that βk is an element of class h([β |Skn ]) for k < K and β |∨e Sn = γ |∨e Sn when γ ∈ f ([α |∨e Sn ]). k≥K K k≥K K k≥K K By [1, Lemma 2.2], β is continuous and ψ is an isomorphism. Corollary 2.5. Let X and Y be freely homotopy equivalent and (X,x0 ) and (Y,y0 ) are semi-locally pointed homotopy equivalent, then Hn (X,x0 ) ∼ = Hn (Y,y0 ). Proof. It is known that two free homotopy equivalent spaces have isomorphic homotopy groups, so by Theorem 2.4 the result holds. Example 2.6. Let CHn be the cone over the n-dimensional Hawaiian Earring. By [1], we can conclude that Hn (CHn , θ ) ∼ ̸ Hn (CHn ,∗). = So there is no pointed homotopy equivalent neighbourhoods of θ and ∗. Corollary 2.7. If (X,x0 ) and (Y,y0 ) are semi-locally pointed homotopy equivalent, then Hn (CX, x˜t ) ∼ = Hn (CY, y˜t ), in which x˜t = [x0 ,t] and y˜t = [y0 ,t ′ ], t,t ′ ̸= 1. Proof. We know that the cone over any space X is contractible, and so πn (CX) is trivial. Consequently, Theorem 2.4 gives the result.

36


Corollary 2.8. Suppose x0 ,x1 are two points of space X with a path from x0 to x1 , so that x0 and x1 have pointed homotopy equivalent open neighbourhoods. Then Hn (X,x0 ) ∼ = Hn (X,x1 ). Proof. We know that if X is path connected, then πn (X,x0 ) ∼ = πn (X,x1 ) for all distinct points x0 ,x1 ∈ X. So by Theorem 2.4, we conclude the result. Remark 2.9. As an example, if X is a self-similar path connected space such as Seipeińsky gasket or Menger sponge, then for all n ∈ N, Hn (X) is independent of the choice of the base point.

Seipeińsky gasket

Menger sponge

References [1] A. Babaee, B. Mashayekhy and H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topology Appl. 159 (2012), no. 8, 2043–2051. [2] K. Eda and K. Kawamura, Homotopy and homology groups of the n dimensional Hawaiian Earring, Fundamenta Mathematicae 165 (2000), no. 1, 17–28. [3] U.H. Karimov and D. Repovˇs, Hawaiian groups of topological spaces (Russian), Uspekhi. Mat. Nauk. 61 (2006), no. 5 (371), 185–186; transl. in Russian Math. Surv. 61 (2006), no. 5, 987–989.

37

B. Bidabad and M. Yarahmadi :: Ricci flow and convergence of evolving Finslerian metrics

Paper No. 1.8

Ricci flow and convergence of evolving Finslerian metrics Behroz Bidabad1 and Mohamad Yarahmadi2 e-mail: [email protected], Amirkabir University of Technology, Tehran, Iran. 2 e-mail: [email protected], Amirkabir University of Technology, Tehran, Iran. 1

Abstract Here the convergence of evolving Finslerian metrics is studied. More intuitively it is proved that a family of Finslerian metrics g(t) which are solutions of Ricci flow converge in C∞ to a smooth limit Finslerian metric g¯ as t approaches the finite time T . Keywords: Finsler geometry, Ricci flow,

1 Introduction Ricci flow is a branch of general geometric flows, which is an evolution equation for a Riemannian metric. Ricci flow can be used to deform an arbitrary metric into a metric, from which one can determine the topology of the underlying manifold. Hence Ricci flow innovate numerous progress in the proof of many geometric conjectures. Geometric flows are not only applied in physics and mechanics but also has many real world applications, for instance in the problem of 3-dimensional face recognition in computer science. In 1982 Hamilton introduced the notion of Ricci flow on Riemannian manifolds by ∂ gi j = −2Rici j , g(t = 0) := g0 . ∂t The Ricci flow, which evolves a Riemannian metric by its Ricci curvature is a natural analogue of the heat equation for metrics. In 1989 W.X. Shi determined estimates for the covariant derivatives of the curvature tensor on complete Riemannian manifolds [5]. The derivative estimates established by Shi enable one to prove the long time existance theorem for the flow. More precisely it states that a unique solution to the Ricci flow exists as long as its curvature remains bounded. R. Hamilton [3] has shown that any solution to the Ricci flow which develops a singularity in finite time must have unbounded curvature tensor. Moreover N. Sesum proved that any solution to the Ricci flow which develops a singularity in finite time must have unbounded Ricci curvature [4]. These estimates play an important role to show that the Riemannian metric on S2 converges to a constant curvature metric. The derivative estimates are used to prove Hamiltons convergence criterion for the Ricci flow. In the present work, we prove that a family of Finslerian metrics g(t) of the solution converge in C∞ to a smooth limit Finslerian metric g¯ as t approaches T .

38

∂ ∂t g(t)

= ω(t)


In our future works we are going to use this result to derive evolution equations for the hh− curvature tensor and its covariant derivatives. It will estimate the covariant derivatives of Cartan curvature tensor. Next as an application, one can prove that a family of solutions of the compact Finslerian Ricci flow g(t) converge in C∞ to a smooth limit Finslerian metric g¯ as t approaches T . Finally, the later result may be used to show that the compact Finsler Ricci flow cannot develop a singularity in finite time unless the hh−curvature is bounded. These tools are indispensable for development of Ricci flow on Finsler geometry. specially for determining interior estimation of some solutions.

2 Preliminaries Let M be an n-dimensional C∞ manifold, we denote by T M its tangent bundle and by π : T M0 −→ M, fiber bundle of nonzero tangent vectors. A Finsler structure on M is a function F : T M −→ [0,∞) with the following properties : (1) Regularity: F is C∞ on the entire slit tangent bundle T M0 = T M\0; (2) Positive homogeneity: F(x,λ y) = λ F(x,y) ∀λ > 0; (3) Strong convexity: The n × n Hessian matrix (gi j ) = 12 ([F 2 ]yi y j ) is positive-definite at every point of T M\0.The pair (M,F) is said to be a Finsler ˆ , and X˙ = µ(X), ˆ where ρ and µ are morphisms manifold. Let Xˆ ∈ T T M0 be a complete lift of X = ρ(X) ˆ =∇ˆv defined by ρ : T T M0 → π ∗ T M, ρz (( δδxi )z ) = ( ∂∂xi )z , ρ(( ∂∂yi )z ) = 0 and µz : Tz T M0 → Tπz M, µz (X) X and v = vi ∂∂xi . Recall that { δδxi , ∂∂yi } are a basis for T T M , where ∂ 2y j

δ δ xi

=

∂ ∂ xi

j

− Gi ∂∂y j , Gij =

∂ Gi ∂yj

and

∂ F2

Gi = 41 gih ( ∂ yh ∂ x j − ∂ xh ). The connection ∇ is said to be regular, if µ defines an isomorphism between V T M0 and π ∗ T M. In this case, there is the horizontal distribution HT M such that we have the Whitney sum T T M0 = HT M ⊕V T M. It can be shown that the set { δδx j } and { ∂∂y j }, forms a local frame field for the horizontal and vertical subspaces, respectively. This decomposition permits to write a vector Xˆ ∈ T T M0 into the form Xˆ = ˆ = ρ(Yˆ ), and the H Xˆ +V Xˆ uniquely. In the sequel, we denote all the sections of π ∗ T M by X = ρ(X),Y ˆ Yˆ respectively, unless otherwise specified. The torsion tensor corresponding complete lift on T M0 by X, of the Finsler connection ∇ is defined by ˆ Yˆ ) = ∇ ˆ Y − ∇ ˆ X − ρ[X, ˆ Yˆ ]. τ(X, X Y ˙ ) = τ(V X,H ˆ Yˆ ) and S(X,Y ) = τ(H X,H ˆ Yˆ ) where Which determines two torsion tensors defined by T (X,Y H Xˆ ∈ Hz T M = ker µz and V Xˆ ∈ Vz T M = kerπ∗z , and π∗z is the tangent mapping of the canonical projection π. There is a unique regular connection ∇ relative to the Finsler structure F such that ∇Zˆ g = 0, ˙ ),Z) = g(T (X,Z),Y ˙ S(X,Y ) = 0 and g(T (X,Y ), called Cartan connection. The Cartan connection on π ∗ T M is given as follows ∇ : Tz T M0 × Γ(π ∗ T M) −→ Γ(π ∗ T M). ˆ ) −→ ∇ ˆ Y (X,Y X

39


The linear connection ∇ satisfies ˆ ˆ ˆ Yˆ ),Z) + g(τ(Z, ˆ X),Y ˆ ) + g(τ(Z, ˆ Yˆ ),X) 2g(∇Xˆ Y,Z) =X.g(Y,Z) + Yˆ .g(X,Z) − Z.g(X,Y ) + g(τ(X, ˆ Yˆ ],Z) + g(ρ[Z, ˆ X],Y ˆ ) + g(ρ[Z, ˆ Yˆ ],X). + g(ρ[X, (2.1) Here, we denote by ∇m A the mth iterated covariant derivative of the tensor A. Given two tensors A and B we shall write A ? B for any bilinear expression in A and B.

3 Main results Let M be a compact manifold, F(t) a smooth one parameter family of Finsler structures and g(t) the Hessian matrix of F(t). Suppose that ∂t∂ g(t) = ω(t), where ω(t) is family of symmetric (0,2)−tensors. We define a function u0 : [0,T ) −→ R by u0 (t) = supT M |ω(t)|F(t) , for all t ∈ [0,T ). Lemma 3.1. Assume that 0T u0 (t)dt < ∞. Then the Finslerian metrics are uniformly equivalent; that is, there exists a positive constant C such that R

1 2 |v| ≤ |v|2F(t) ≤ C|v|2F(0) , C F(0) for all points ((p,x),t) ∈ T M × [0,T ) and all vectors v ∈ Tp M. Proof. Fix a point ((p,x),t) ∈ T M × [0,T ) and a vector v ∈ Tp M. Then d | |v|2F(t) | ≤ |ω(t)|F(t) |v|2F(t) ≤ u0 (t)|v|2F(t) , dt since

RT 0

u0 (t)dt < ∞, the assertion follows.

The following proposition allows us to relate the time dependent connection ∇ to a fixed background connection D. Proposition 3.2. Let (M,F) be a Finsler space, and ∇ the Cartan connection associated with the fundamental function F and g the Hessian matrix of F. Moreover, suppose that D is a torsion free connection. Then, ˆ Yˆ ), ∇Xˆ Y = DXˆ Y + Γ(X, where ˆ Yˆ ),Z) =(D ˆ g)(Y,Z) + (D ˆ g)(X,Z) − (D ˆ g)(X,Y ) 2g(Γ(X, X Y Z ˆ ˆ ˆ ˆ ˆ Yˆ ),X). + g(τ(X, Y ),Z) + g(τ(Z, X),Y ) + g(τ(Z,

40


ˆ Yˆ ] = D ˆ Y − D ˆ X and by putting it in relation 2.1 we have Proof. Since D is torsion free, that is ρ[X, X Y ˆ ˆ ˆ Yˆ ),Z) + g(τ(Z, ˆ X),Y ˆ ) + g(τ(Z, ˆ Yˆ ),X) + Yˆ .g(X,Z) − Z.g(X,Y ) + g(τ(X, 2g(∇Xˆ Y,Z) =X.g(Y,Z) + g(DXˆ Y − DYˆ X,Z) + g(DZˆ X − DXˆ Z,Y ) + g(DZˆ Y − DYˆ Z,X). Therefore 2g(∇Xˆ Y,Z) =(DXˆ g)(Y,Z) + (DYˆ g)(X,Z) − (DZˆ g)(X,Y ) ˆ Yˆ ),Z) + g(τ(Z, ˆ X),Y ˆ ) + g(τ(Z, ˆ Yˆ ),X) + 2g(D ˆ Y,Z). + g(τ(X, X

Now by using definition of Γ we conclude that ˆ Yˆ ),Z) + 2g(D ˆ Y,Z). 2g(∇Xˆ Y,Z) =2g(Γ(X, X Finally we have ˆ Yˆ ) + D ˆ Y,Z). g(∇Xˆ Y,Z) =g(Γ(X, X Therefore the assertion follows. Lemma 3.3. Let ∇ denote the Cartan connection associated with the metric g(t) and D a fixed background a torsion free connection. Then m−1

∇m ω(t) − Dm ω(t) = ∑ l=0

Di1 g(t) ? ... ? Diq g(t) ? Dl ω(t)

∑

q ∑ j=1 i j =m−l

m−1

+

∑ l=0

Di1 g(t) ? ... ? Diq g(t) ? Dl ω(t),

∑

q ∑ j=1 i j =m−l−1

for m = 1,2,.... Proof. The proof is by induction on m. For each integer m ≥ 1, we define continuous functions um : [0,T ) −→ R and uˆm : [0,T ) −→ R by um (t) = sup | ∇m ω(t) |F(t) ,

uˆm (t) = sup | Dm ω(t) |F(0) ,

TM

TM

for each t ∈ [0,T ). By means of the following identity Zt

g(t) = g(0) +

ω(τ)dτ,

(3.1)

0

we obtain m

sup | D g(t) |F(0) ≤ TM

for all t ∈ [0,T ).

41

Zt 0

uˆm (τ)dτ,

(3.2)


Lemma 3.4. Suppose that

RT 0

um (t)dt < ∞ for m = 0,1,2,.... Then

RT 0

uˆm (t)dt < ∞ for m = 0,1,2,....

Proof. The proof is by induction on m. Fix an integer m ≥ 1, and suppose that 0T uˆl (t)dt < ∞ for l = 0,1,2,...,m − 1. It follows from 3.2 that sup sup |Dl g(t)|F(0) < ∞, for l = 1,2,...,m − 1. Moreover, R

t∈[0,T ) T M

the metrics g(t) are uniformly equivalent by Lemma 3.1. Using Lemma 3.3, we obtain m−1

|Dm ω(t)|F(0) − |∇m ω(t)|F(0) ≤C1

∑

|Di1 g(t)|F(0) ...|Diq g(t)|F(0) |Dl ω(t)|F(0)

∑

l=0 i1 +...+iq =m−l m−1

+C2

∑

∑

|Di1 g(t)|F(0) ...|Diq g(t)|F(0) |Dl ω(t) |F(0) ,

l=0 i1 +...+iq =m−l−1

for some positive constant C1 and C2 . This implies m−1

|Dm ω(t)|F(0) ≤|∇m ω(t)|F(0) +C3

∑ |Dl ω(t)|F(0) +C3 1 + |Dmg(t)|F(0)

|ω(t)|F(0) .

l=1

Therefore m−1

|Dm ω(t)|F(0) ≤ C4 |Dm ω(t)|F(0) +C3

∑ |Dl ω(t)|F(0) +C3C4 1 + |Dmg(t)|F(0)

|ω(t)|F(0) .

l=1

Using 3.2, we have Zt uˆm (t) ≤ C4 um (t) +C3 ∑ uˆl (t) +C3C4 u0 (t) 1 + uˆm (τ)dτ , m−1 l=1

0

for all t ∈ [0,T ). This implies Zt

m−1 d log 1 + uˆm (τ)dτ ≤ C4 um (t) +C3 ∑ uˆl (t) +C3C4 u0 (t), dt l=1 0

for all t ∈ [0,T ). By assumption, we have 0T u0 (t)dt < ∞ and 0T um (t)dt < ∞. Moreover, the induction R hypothesis implies that 0T uˆl (t)dt < ∞ for l = 1,2,...,m − 1. Putting these facts together, we conclude R that 0T uˆm (τ)dτ < ∞. R

R

Theorem 3.5. Suppose that 0T um (t)dt < ∞ for m = 0,1,2,.... Then the metrics g(t) converge in C∞ to a smooth limit metric g¯ when t −→ T . R

Proof. By last Lemma, we have 0T uˆm (τ)dτ < ∞ for m = 1,2,.... therefore by relation 3.1, the metrics g(t) converge in C∞ to a symmetric (0,2)−tensor g.¯ Moreover, it follows from Lemma 3.1 that g¯ is positive definite. R

42


References [1] B. Bidabad, M.Yarahmadi, On quasi-Eienstein Finsler spaces, Accepted in Bulletin of the Iranian Mathematical Society, 2013. [2] B. Bidabad, M.Yarahmadi, Evolution of the Cartan curvature tensor on Finsler spaces, The 44th Annual Iranian Mathematics Conference, 27-30 August 2013. [3] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1989), no. 2, 255-306. [4] N. Sesum, Curvature tensor under the Ricci flow, Amer. J. math. 127, 1315-1324 (2005). [5] W.X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30, 223301 (1989).

43

M. Nadjafikhah and M. Hesamiarshad :: The equivalence problem for equations of the form uxxx = ut + A(t, x)ux + B(t, x)u + C(t, x)

Paper No. 1.9

the equivalence problem for equations of the form uxxx = ut + A(t, x)ux + B(t, x)u +C(t, x) 1

Mehdi Nadjafikhah1 and Mostafa Hesamiarshad2 e-mail: m [email protected], Karaj Branch,Islamic Azad University, Karaj, Iran. 2 e-mail: [email protected], Karaj Branch,Islamic Azad University, Karaj, Iran. Abstract

The moving coframe method is applied to solve the local equivalence problem for equations of the form uxxx = ut + A(t, x)ux + B(t, x)u +C(t, x) in two independent variables under action of a pseudogroup of contact transformations to determine necessary and sufficient conditions for this equations to be equivalent to simplest model under a contact transformation. Keywords: Contact transformations , Equivalence problem, Moving coframe.

Introduction In the early twentieth century, Elie Cartan developed a uniform method for analyzing the differential invariants of many geometric structures, nowadays called the ‘Cartan equivalence method’. Also, the method of equivalence is a systematic procedure that allows one to decide whether two systems of differential equations can be mapped one to another by a transformation taken in a given pseudo-group. Later, C. Erhesmann and S. Chern introduced two important concepts to the method of equivalence: jets spaces and G-structures. In recent years, thanks to mathematical software, many authors have successfully applied the method of equivalence to many interesting problems such as classifications of differential equations (Kamran, Olver and Fels), holonomy groups (Bryant), inverse variational problems (Fels) and general relativity (Newman). In this paper, we consider a local equivalence problem for the equations uxxx = ut + A(t, x)ux + B(t, x)u +C(t, x)

(0.1)

under the contact transformation of a pseudo-group. Two equations are said to be equivalent if there exists a contact transformation maps one equation to another.We use Elie Cartan’s method of equivalence [1, 6, 7] in its form developed by Fels and Olver [2, 3] and as stated by morozov [4] to compute the Maurer - Cartan forms, structure equations, basic invariants, and the invariant derivatives for symmetry groups of equations. Cartan’s solution to the equivalence problem states that two equations are (locally) equivalent if and only if Cartan’s test satisfied and essential torsion coefficients in the structure equations are constant or their classifying manifolds (locally) overlap. The symmetry classification problem for classes of differential equations is closely related to the problem of local equivalence: symmetry groups

44


of two equations are necessarily isomorphic if these equations are equivalent,while the converse statement is not true in general. For the symmetry analysis of (0.1) the reader is referred to [5].

1 Pseudo-group of contact transformations and symmetries of differential equations In this section we describe the local equivalence problem for differential equations under the action of the pseudo group of contact transformations. Two equations are said to be equivalent if there exists a contact transformation which maps the equations to each other. We apply Elie Cartan’s structure theory of Lie pseudo-groups to obtain necessary and sufficient conditions under which equivalence mappings can be found. This theory describes a Lie pseudo-group in terms of a set of invariant differential 1-forms called Maurer-Cartan forms, which contain all information about the seudo-group. In particular, they give basic invariants and operators of invariant differential, which in terms allow us to solve equivalence problem for submanifolds under the action of the pseudo-group. Recall that expansions of exterior differentials of Maurer-Cartan forms in terms of the form themselves, yields the Cartan structure equation for the prescribed pseudo-group. Suppose π : Rn × Rm → Rn is a trivial bundle with the local base coordinates (x1 ,... ,xn ) and the local fibre coordinates (u1 ,... ,um ); then J 1 (π ) is denoted by the bundle of the firstorder jets of sections of π , with the local coordinates (xi , uα , pαi ), i ∈ {1,... ,n}, α ∈ {1,... ,m}, where pαi = ∂∂uxαi . For every local section (xi , fα (x)) of π , the corresponding 1-jet (xi , fα (x), ∂ f∂αx(x) ) is denoted i 1 by j1 ( f ). A differential 1-form ν on J (π ) is called a contact form, if it is annihilated by all 1-jets of local sections: j1 ( f )∗ ν = 0. In the local coordinates every contact 1-form is a linear combination of the forms ν α = duα − pαi dxi , α ∈ {1,... ,m}. A local diffeomorphism: ∆ : (xi , uα , pαi ) → (xi , uα , pαi )

∆ : J 1 (π ) → J 1 (π ),

is called a contact transformation, if for every contact 1-form ν , the form ∆∗ ν is also contact. It was shown in [4] that the following differential 1-forms, β

Θα = aαβ (duβ − p j dx j ), Ξi = bij dx j + ciβ Θβ , β

Σαi = aαβ Bij d p j + fiαβ Θβ + gαi j Ξ j . are the Maurer-Cartan forms of Cont(J 1 (π )). They are defined on J 1 (π ) × H , where H = (aαβ , bij , ciβ , fiαβ , gαi j ) | α , β ∈ {1,... ,m}, i, j ∈ {1,... ,n}, det(aαβ ).det(bij ) 6= 0, gαi j = gαji and (Bij ) is

45


the inverse matrix for (bij ). They satisfy the structure equations dΘα = Φαβ ∧ Θβ + Ξk ∧ Σαk , dΞi = Ψik ∧ Ξk + Πiγ ∧ Θγ , γ

dΣαi = Φαγ ∧ Σi − Ψki ∧ Σαk + Λαiβ ∧ Θβ + Ωαi j ∧ Ξ j . where the forms Φαβ , Ψij , Πiβ , Λαiβ and Ωαi j depend on differentials of the coordinates of H . Suppose R is a first-order differential equation in m dependent and n independent variables. We consider R as a subbundle in J 1 (π ). Suppose Cont(R) is the group of contact symmetries for R. It consists of all the contact transformations on J 1 (π ) mapping R to itself. Differential equations defines a submanifold R ⊂ J 1 (π ). The Maurer-Cartan forms for its symmetry pseudo-group Cont(R) can be found from restrictions θ α = ı∗ Θα , ξ i = ı∗ Ξi and σiα = ı∗ Σαi , where ı = ı0 ×id : R ×H −→ J 1 (π )×H with ı0 : R −→ J 1 (π ) is defined by our differential equations. In order to compute the Maurer-Cartan forms for the symmetry pseudogroup, we implement Cartan’s equivalence method. Firstly, the forms θ α , ξ i , σiα are linearly dependent, i.e. there exists a nontrivial set of functions Uα ,Vi ,Wαi on R × H such that Uα θ α +Vi ξ i +Wαi σiα ≡ 0. Setting these functions equal to some appropriate constants allows us to introduce a part of the coordinates of H as functions of the other coordinates of R × H . Secondly, we substitute the obtained values into γ the forms φβα = ı∗ Φαβ and ψki = ı∗ ψki coefficients of semi-basic forms φβα at σ j , ξ j , and the coefficients of γ

semi-basic forms ψ ij at σ j are lifted invariants of Cont(R). We set them equal to appropriate constants and get expressions for the next part of the coordinates of H , as functions of the other coordinates of R × H . Thirdly, we analyze the reduced structure equations d θ α = φβα ∧ θ β + ξ k ∧ σkα , d ξ i = ψki ∧ ξ k + πγi ∧ θ γ , γ

d σiα = φγα ∧ σi − ψik ∧ σkα + λiαβ ∧ θ β + ωiαj ∧ ξ j . If the essential torsion coefficients are dependent on the group parameters , then we may normalize them to constants and find some new part of the group parameters, which, upon being substituted into the reduced modified Maurer-Cartan forms, allows us to repeat the procedure of normalization. This process has tow results. First, when the reduced lifted coframe appears to be involutive, this coframe is the desired set of defining forms for Cont(R). Second, when the coframe is not involutive we should apply the procedure of prolongation described in [6].

2 Structure and invariants of symmetry groups for equations of the form uxxx = ut + A(t, x)ux + B(t, x)u +C(t, x) Consider the following system equivalent to (0.1) of first order: ux = v

vx = w,

wx = ut + A(t, x)ux + B(t, x)u +C(t, x)

46

(2.1)


We apply the method described in the previous section to the class of equations (2.1). We denotes that t = x1 , x = x2 , u = u1 , v = u2 , w = u3 , ut = p11 , ux = p12 , vt = p21 , vx = p22 , wt = p31 , wx = p32 . We consider this system as a sub-bundle of the bundle J 1 (π ), π : R2 × R3 −→ R3 , with local coordinates {x1 , x2 , u1 , u2 , u3 , p11 , p21 , p31 }, where the embedding ι is defined by the equalities: p12 = u2

p22 = u3

p32 = p11 + A(x1 , x2 )u2 + B(x1 , x2 )u1 +C(x1 , x2 )

(2.2)

The forms θ α = ι ∗ Θα , α ∈ {1, 2, 3}, ξ i = ι ∗ Ξi , i ∈ {1, 2}, are linearly dependent. The group parameters aαβ , bij should satisfy the simultaneous conditions det(aαβ ) 6= 0, det(bij ) 6= 0. Linear dependence between the forms σiα are σ21 = 0, σ22 = 0, σ23 = σ11 (2.3) 1 , f1 , f1 , Computing the linear dependence conditions (2.3) gives the group parameters a11 , a12 , a13 , a23 , b12 , f11 21 22 1 , f 2 , f 2 , f 2 , f 3 , f 3 , g1 , g1 , g2 , g2 , g3 , g3 as functions of other group parameters and the local cof23 21 22 23 22 23 12 22 12 22 12 22 ordinates {x1 , x2 , u1 , u2 , u3 , p11 , p21 , p31 } of R1 .In particular, 2

2 f21

=−

g212 =

2

2

2

a21 a32 a22 − (a21 ) a33 − a31 (a22 ) + g222 c21 (a33 ) b11 a22 + g212 c11 (a33 ) b11 a22 2

(a33 ) b11 a22

−p31 a22 b22 − p21 a21 b22 + (Au2 + Bu1 +C)b21 a22 + p11 b21 a22 + u3 b21 a21

2 b11 (b22 ) a3 a2 + a2 a3 + g2 c2 a2 b2 a3 + g2 c1 a2 b2 a3 2 f22 = − 2 2 1 3 22 32 22 22 3 12 2 2 2 3 , a3 b2 a2 (Au2 + Bu1 +C)a22 + p11 a22 + u3 a21 g222 = − , 2 (b22 ) a2 + g2 c2 a3 b2 + g2 c1 a3 b2 2 f23 = − 2 22 3 3 3 2 2 12 3 3 2 , a3 b2 2 2 a33 b11 + g122 c22 a22 (b22 ) + g112 c12 a22 (b22 ) 1 f22 = − , 2 a22 (b22 ) a2 − g1 c2 a2 b2 − g1 c1 a2 b2 1 f21 = 1 22 1 2 2 2 2 12 1 2 2 , a2 b2 3 2 a (u3 b1 − p21 b22 ) g112 = 3 , 2 (b22 )

, b12 = 0,

,

a23 = 0, a13 = 0, a11 =

a33 b11 , b22

1 f23 = −g122 c23 − g112 c13 ,

a12 = 0, g122 =

u3 a33 b11 2

(b22 )

.

1 , f 3 , f 3 , g3 and g3 are too long to be written out here completely. The expressions for f11 22 22 23 12 The analysis of the semi-basic modified Maurer-Cartan forms φβα , ψki at the obtained values of the group parameters gives the following normalizations. The form ψ21 is semi-basic, and ψ21 ≡ −c13 σ11 . So we take c13 = 0. For the semi-basic form φ21 we have

φ21 ≡ −c12 σ11 (mod θ 1 , θ 2 , θ 3 , ξ 1 , ξ 2 ),

47


thus, we can assume c12 = 0. And for the semi-basic form φ31 we have

φ31 ≡ −

1 (b2 )3 c23 a33 b21 u3 − c23 a33 b22 p21 + f13 2 ξ 1 (mod θ 1 , θ 2 , θ 3 ), 2 3 (b2 )

so we set the coefficient at ξ 1 equal to 0 and find 1 f13

c23 a33 (b22 p21 − b21 u3 ) . = (b22 )3

Doing the analysis of the modified semi-basic Maurer-Cartan forms in the same way, we can normalize the following group parameters: a21 = a31 = a32 = 0,

a22 = a33 b22 ,

3

b11 = (b22 ) , c11 = c12 = c13 = c21 = c22 = c23 = 0, 1 2 2 2 3 3 3 f12 = f11 = f12 = f13 = f21 = f12 = f11 = 0,

b21 = −Ab22 , 3 2 f13 = f12 =

B 3

(b22 )

.

At the third step, the structure equations of the symmetry group for system (2.1) have the form: d θ 1 = (η4 + 2η5 ) ∧ θ 1 + 2I θ 1 ∧ ξ 1 − θ 2 ∧ ξ 2 + ξ 1 ∧ σ23 , d θ 2 = (η4 + η5 ) ∧ θ 2 + I θ 2 ∧ ξ 1 − θ 3 ∧ ξ 2 + ξ 1 ∧ σ12 , d θ 3 = η4 ∧ θ 3 + ξ 1 ∧ σ13 + ξ 2 ∧ σ23 , d ξ 1 = 3 η5 ∧ ξ 1 , d ξ 2 = η5 ∧ ξ 2 , d σ12 = η1 ∧ ξ 1 + ξ 2 ∧ σ13 + (η4 − 2η5 ) ∧ σ12 − J θ 2 ∧ ξ 2 − I θ 3 ∧ ξ 2 , d σ13 = η2 ∧ ξ 1 + η3 ∧ ξ 2 + (η4 − 3η5 ) ∧ σ13 , d σ23 = η3 ∧ ξ 1 + (η4 − η5 ) ∧ σ23 − J θ 1 ∧ ξ 2 − I θ 2 ∧ ξ 2 + J θ 3 ∧ ξ 1 +ξ 2 ∧ σ12 ,

(2.4)

where I = ∂∂Ax , J = ∂∂Bx . If I = 0 and J = 0, we can’t absorb any group parameters more than before. Besides, the Cartan character is s1 = 5 and the indetermination degree is 3, thus the involution test fails. Therefore we use the procedures of prolongation to compute the new structure equations: d θ 1 = −θ 1 ∧ (η4 + 2η5 ) − θ 2 ∧ ξ 2 + ξ 1 ∧ σ11 , d θ 2 = −θ 2 ∧ (η4 + η5 ) − θ 3 ∧ ξ 2 + ξ 1 ∧ σ12 , d θ 3 = −θ 3 ∧ η4 + ξ 1 ∧ σ13 + ξ 2 ∧ σ11 , d ξ 1 = −3ξ 1 ∧ η5 , d ξ 2 = −ξ 2 ∧ η5 ,

48

(2.5)


d σ11 = −ξ 1 ∧ η3 + ξ 2 ∧ σ12 − σ11 ∧ (η4 − η5 ), d σ12 = −ξ 1 ∧ η1 + ξ 2 ∧ σ13 − σ12 ∧ (η4 − 2η5 ), d σ13 = −ξ 1 ∧ η2 − ξ 2 ∧ η3 + σ13 ∧ (η4 − 3η5 ), d η1 = −β1 ∧ ξ 1 + ξ 2 ∧ η2 − η1 ∧ (η4 − 5η5 ), d η2 = −β2 ∧ ξ 1 − β3 ∧ ξ 2 − η2 ∧ (η4 − 6η5 ), d η3 = −β3 ∧ ξ 1 + ξ 2 ∧ η1 − η3 ∧ (η4 − 4η5 ), d η4 = 0, d η5 = 0. In structure equations (2.5), the forms η1 , ··· η5 on J 2 (π ) × H depend on differentials of the parameters of H , while the forms β1 , β2 , β3 depend on differentials of the prolongation variables. In the structure equations (2.5) the degree of indetermination is 3 and the Cartan characters are s1 = 3, s2 = ... = s13 = 0. Consequently, Cartan’s test for the lifted coframe {θ 1 , θ 2 , θ 3 , ξ 1 , ξ 2 , σ11 , σ12 , σ13 , η1 , η2 , η3 , η4 , η5 } is satisfied. Therefore, the coframe is involutive. All the essential torsion coefficients in the structure equations (2.5) are constant. By applying Theorem 11.8 and Theorem 8.16 of [6], we have: Theorem 2.1. Equations of the form uxxx = ut + A(t)ux + B(t)u + C(t, x) are equivalent to the equation uxxx = ut under a contact transformation.

References [1] E.Cartan, Les Problemes d’equivalence, Oeuvres Completes Vol. 2, Gauthiers-Villars, Paris, 1953. [2] M.Fels, P.J.Olver, 1998, Moving coframes, I. A practical algorithm, Acta. Appl. Math 51 161-213. [3] M.Fels, P.J.Olver, 1999 Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math 55127-208. [4] O.Morozov, Moving coframes and symmetries of differential equations, J. Phys. A: Math. Gen. 35 (2002) 2965-2977. [5] F. Gungor and V.I.Lahno and R.Z.Zhdanov, Symmetry classification of KdV-type nonlinear evolution equations, journal of mathematical phisycs,volume 45,number 6, June 2004. [6] P.J.Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, 1995. [7] R.B.Gardner, The Method of Equivalence and Its Applications, SIAM, Philadelphia, 1989.

49

B. Bidabad and M. Sedaghat :: Extrinsic sphere in Finsler geometry

Paper No. 1.10

Extrinsic sphere in Finsler geometry Behroz Bidabad1 and Maral sedaghat2 e-mail: [email protected], Amirkabir University of Technology, Tehran, Iran. 2 e-mail: m [email protected], Amirkabir University of Technology, Tehran, Iran. 1

Abstract Here, the notion of extrinsic sphere in Finsler geometry is defined. Next it is shown that a submanifold S of a Finsler manifold M is an extrinsic sphere if and only if every circle in S is a circle in M. Keywords: Finsler, mean curvature vector, totally umbilical, extrinsic sphere.

1 Introduction An n-dimensional Riemannian manifold is called an intrinsic sphere if it is locally isometric to an ordinary sphere in a Euclidean space. An n(≥ 2)-dimensional submanifold of an arbitrary Riemannian manifold is said to be an extrinsic sphere if it is totally umbilical and has non-zero parallel mean curvature vector. In this situation, it is well known that an extrinsic sphere in a Euclidean space is an intrinsic sphere. However, in genaral, an extrinsic sphere is not always an intrinsic sphere, namely, an extrinsic sphere is not always isometric with a sphere. Since the definition of extrinsic sphere is based on the extrinsic properties of submanifold, we shall use the terminology of extrinsic sphere. We should notice that these assupmtions do not imply topological restrictions on the submanifold in the following sence: ˜ it is always posible to find a metric on M˜ such that M If M is a submanifold of a Riemannian manifold M, is an extrinsic sphere. Extrinsic spheres in Riemannian geometry have been geometrically characterized by Nomizu and Yano, cf., [6]. It has been shown that a submanifold M of a Riemannian manifold M˜ is an extrinsic sphere if and only if every curve τ on M starting at a point x is developed upon a curve ˜ The notion of development is due to lying in a certain Euclidean n-sphere in the tangent space Tx M. Cartan. By means of the concept of extrinsic sphere and based on the axiom of r-planes in Riemannian geometry defined by E. Cartan, Leung and Nomizu proposed the axiom of r-spheres and proved that if a Riemannian manifold M of dimension n ≥ 3 satisfies the axiom of r-spheres for some r, 2 < r < n, then M has constant sectional curvature. In the present note a natuaral analogue definition of extrinsic sphere in Finsler geometry is given. Next it is proved that a submanifold S of a Finsler manifold M is an extrinsic sphere if and only if every circle in S is a circle in M.

50


2 Preliminaries Let M be a real n-dimensional manifold of class C∞ . We denote by T M the tangent bundle of tangent vectors, by p : T M0 −→ M the fiber bundle of non-zero tangent vectors and by p∗ T M −→ T M0 the pulled-back tangent bundle. Let (x,U) be a local chart on M and (xi ,yi ) be the induced local coordinates on p−1 (U). A Finsler structure on M is a function F : T M −→ [0,∞), with the following properties:(i) F is differentiable C∞ on T M0 ; (ii) F is positively homogeneous of degree one in y, that is, F(x, λ y) = λ F(x,y), for all λ > 0; (iii) The Finsler metric tensor g defined by the Hessian matrix 2 of F 2 , (gi j ) = ( 12 [ ∂ y∂i ∂ y j F 2 ]), is positive definite on T M0 . A Finsler manifold is a pair (M,F) consisting of a differentiable manifold M and a Finsler structure F on M. Here and everywhere in this paper all the indices i, j,...,a,b,...α , β ,... run over the range 1,...,n. Any point of T M0 is denoted by z = (x,y), where x = pz ∈ M and y ∈ Tp(z) M. We denote by T T M0 , the tangent bundle of T M0 and by ρ , the canonical linear mapping ρ : T T M0 −→ p∗ T M, where, ρ = p∗ . For all z ∈ T M0 , let Vz T M be the set of vertical vectors at z, that is, the set of vectors which are tangent to the fiber through z. Equivalently, Vz T M = ker p∗ where p∗ : T T M0 −→ T M is the linear tangent mapping. Let ∇ be a linear connection on ˆ = ∇ˆy the vector bundle p∗ T M −→ T M0 . Consider the linear mapping µ : T T M0 −→ p∗ T M, by µ (X) X ∗ where, Xˆ ∈ T T M0 and y is the canonical section of p T M. The connection ∇ is said to be regular, if µ defines an isomorphism between V T M0 and π ∗ T M. In this case, there is the horizontal distribution HT M such that we have the Whitney sum T T M0 = HT M ⊕V T M. This decomposition permits to write a vector field Xˆ ∈ χ (T M0 ) into the horizontal and vertical form Xˆ = H Xˆ +V Xˆ uniquely. In the sequel, we ˆ Yˆ , etcetera and the corresponding sections of π ∗ T M by X = ρ (X), ˆ denote all vector fields on T M0 by X, Y = ρ (Yˆ ), respectively, unless otherwise specified. The structural equations of the regular connection ∇ are given by: ˆ Yˆ ) = ∇ ˆ Y − ∇ ˆ X − ρ [X, ˆ Yˆ ], τ (X, (2.1) X Y ˆ Yˆ )Z = ∇ ˆ ∇ ˆ Z − ∇ ˆ ∇ ˆ Z − ∇ ˆ ˆ Z, Ω(X, (2.2) X Y

Y X

[X,Y ]

ˆ Y = ρ (Yˆ ), Z = ρ (Z) ˆ and X, ˆ Yˆ and Yˆ are vector fields on T M0 . The torsion tensor τ where, X = ρ (X), and curvature tensor Ω of ∇ determine two torsion tensors denoted here by S and T and three curvature tensors denoted by R, P and Q, defined by: ˆ Yˆ ), T (X,Y ˙ ) = τ (V X,H ˆ Yˆ ), S(X,Y ) = τ (H X,H ˆ Yˆ ), P(X, Y˙ ) = Ω(H X,V ˆ Yˆ ), R(X,Y ) = Ω(H X,H ˙ Y˙ ) = Ω(V X,V ˆ Yˆ ), Q(X, ˆ Y = ρ (Yˆ ), X˙ = µ (X) ˆ and Y˙ = µ (Yˆ ). The tensors R, P and Q are called hh−, hv− and where, X = ρ (X), vv−curvature tensors, respectively. There is a unique regular connection associated with F called Cartan connection such that: ∇Zˆ g = 0, S(X,Y ) = 0, ˆ Yˆ ),Z) = g(τ (V X, ˆ Z),Y ˆ ), g(τ (V X,

51

(2.3)


ˆ Y = ρ (Yˆ ) and Z = ρ (Z), ˆ for all X, ˆ Yˆ , Zˆ ∈ T T M0 , cf., [1]. Let i : S −→ M be a where, X = ρ (X), submanifold of dimension k of the manifold M. We identify any point x ∈ S by its image i(x) and any tangent vector X ∈ Tx S by its image i∗ (X), where i∗ is the linear tangent mapping. Thus Tx S becomes a sub-space of Tx M. Let T S0 nbe the fiber bundle of non-zero tangent vector on S then the canonical injection i induces a mapping i˜ : T S0 −→ T M0 , where z ∈ T S0 identified by i˜(z). Therefore, T S0 is a sub-vector bundle of T M0 and the restriction of p to T S0 is denoted by q : T S0 −→ S. We denote by T¯ (S) = i−1 T M, the induced vector bundle of T M by i. The Finslerian metric of M induces a Finslerian metric on S that we denote it again by g. At a point x = qz ∈ S, where z ∈ T S0 , the orthogonal complement of Tqz S in T¯qz S is denoted by Nqz , namely, T¯x (S) = Tx (S) + Nqz S, where Tx (S) ∩ Nqz S = 0. Denote by P1 : T¯x S −→ Tx S and P2 : T¯x S −→ Nx S, the projection mappings and q−1 T¯ S = q−1 T S + N,

(2.4)

where, N is called the normal fiber bundle. If T T S0 is the tangent vector bundle to T S0 , we denote by ρ , the canonical linear mapping ρ : T T S0 −→ q−1 T S. Let Xˆ and Yˆ be two vector field on T S0 . For z ∈ T S0 , (∇Xˆ Y )z belongs to T¯qz S. Attending to (2.4) we have ˆ ), Y = ρ (Yˆ ), X = ρ (X), ˆ ∇Xˆ Y = ∇¯ Xˆ Y + α (X,Y

(2.5)

where ∇ is the covariant derivative of the Finslerian connection. It results from (2.5) that ∇¯ is a covariant ¯ = 0. α (X,Y ˆ ) derivative in the vector bundle q−1 T S −→ T S0 and is an Euclidean connection, i.e. ∇g belongs to N and is bilinear in Xˆ and Y . Using (2.1), we have ˆ Yˆ ) = P1 τ (X, ˆ Yˆ ) = ∇¯ ˆ Y − ∇¯ ˆ X − ρ [X, ˆ Yˆ ], τ¯ (X, X Y ˆ Yˆ ) = α (X,Y ˆ ) − α (Yˆ ,X), X = ρ (X), ˆ P2 τ (X, Y = ρ (Yˆ ),

(2.6) (2.7)

¯ ∇¯ is said to be the induced connection and α (X,Y ˆ ) the where, τ¯ is the torsion of the connection ∇. 1 ˆ ) is called the mean second fundamental form of the sub-manifold S. The vector field η = n trg α (H X,Y curvature vector field. Definition 2.1. A submanifold of a Finsler manifold is said to be totally umbilical, or simply umbilical, if it is equally curved in all tangent directions. More precisely, let i : S −→ M be an isometric imersion. Then i is called totally umbilical if there exists a normal vector field ξ ∈ N along i such that its second ˆ ) = g(X,Y )ξ for all X,Y ∈ fundamental form α with values in the normal bundle satisfies α (H X,Y −1 ˆ and Y = ρ (Yˆ ). Γ(q T S) where, X = ρ (X) It is well known that if M is an umbilical hypersurface of a Minkowski space (V n+1 ,F) then either M is a Riemannian space or a locally Minkowski space, cf., [5]. Let (V n+1 , α + β ) be a Randers space, where α is a Euclidean metric and β is a closed 1-form, then any complete and connected n-dimensional totally umbilical submanifold of (V n+1 , α + β ) must be either a plane or an Euclidean sphere, cf., [4]. For an example of a totally umbilical submanifold of a Randers spaces one can refer to [4]. Lemma 2.2. Let i : S −→ M be an isometric immersion. If M is totally umbilical then the normal vector field ξ is equal to the mean curvature vector field η .

52


Now, for any Xˆ ∈ χ (T S0 ) and W ∈ Γ(N) we set ∇Xˆ W = −AW Xˆ + ∇¯ ⊥Xˆ W,

(2.8)

where AW Xˆ ∈ Γ(q−1 T S) and ∇¯ ⊥Xˆ W ∈ Γ(N). It follows that ∇¯ ⊥ is a linear connection on the normal bundle N. We also note that A :Γ(N) ⊗ Γ(T T S0 ) −→ Γ(q−1 T S) ˆ = AW Xˆ A(W, X) is a bilinear mapping. For any W ∈ Γ(N) we call AW the shape operator (the Weingarten operator) with respect to W . Finally, (2.8) is named the Weingarten formula for the immersion of S in M. Definition 2.3. A submanifold S of an arbitrary Finsler manifold M is said to be an sphere if it is umbilical and has parallel mean curvature in all direction, that is, for all X ∈ Γ(q−1 T S), we have ∇¯ ⊥H Xˆ η = 0. Since this definition is based on extrinsic properties of the submanifold S, we shall use the terminology of extrinsic sphere. For example, in a Randers space (V n+1 , α + β ), where α is an Euclidean metric and β is a closed 1-form, an extrinsic sphere in V n+1 must be either a plane or an Euclidean sphere after the above mentioned remark on [4]. As another example consider circles as one dimensional extrinsic spheres. A natural definition of a circle in a Finsler manifold is given as follows, cf., [3].

3 Main results Let (M,F) be a Finsler manifold of class C∞ and c : I ⊂ R −→ M a curve parameterized by the arc length s. Let X := c˙ = dc ds be the unitary tangent vector field at each point c(s), cˆ the horizontal lift of c on T S0 and H Xˆ the vector field tangent to c.ˆ c is a circle if there exist a unitary vector field Y = Y (s) along c and a positive constant κ such that ∇H Xˆ X = κY, ∇H Xˆ Y = −κ X, where ∇H Xˆ is the Horizontal Cartan covariant derivative along c. The number κ1 is called the radius of circle. Lemma 3.1. [3] Let c = c(s) be a unit speed curve on an n-dimensional Finsler manifold (M,F). If c is a circle, then it satisfies the following ODE ∇H Xˆ ∇H Xˆ X + g(∇H Xˆ X,∇H Xˆ X)X = 0,

(3.1)

where, g(,) denotes scalar product determined by the tangent vector c.˙ Conversely, if c satisfies (3.1), then it is either a geodesic or a circle.

53


ˆ ) = 0 for any Lemma 3.2. Let S be a submanifold of a Finslerian manifold M. Assume that α (H X,Y −1 orthonormal pair of vectors X,Y ∈ Γ(q T S) at a point x = qz ∈ S. Then the following conclusions hold: ˆ (1) α (H X,X) = α (HYˆ ,Y ) for any orthonormal X,Y ∈ Γ(q−1 T S) at a point x = qz ∈ S. (2) The mean curvature vector ηx is equal to α (H Xˆ1 ,X1 ) where X1 ∈ Γ(q−1 T S) is an arbitrary unit vector field in Tx S. ˆ ) = g(X,Y )ηx for all X,Y ∈ Γ(q−1 T S) at x = qz. (3) S is umbilical at x = qz, i.e. α (H X,Y Now, we are in a position to prove the following theorem, which is a generalization of a result on Riemannian manifolds, cf., [6]. Theorem 3.3. Let S be a connected submanifold of a Finsler manifold M. If for some r > 0, every circle of radius r in S is a circle in M, then S is an extrinsic sphere in M. Conversely, if S is an extrinsic sphere in M, then every circle in S is a circle in M. Proof. To prove the first assertion, let x be an arbitrary point of S and X and Y orthonormal vectors in Tx S. Then there is a circle xs of radius r, | s |< ε , such that 1 x0 = x, X0 = X and (∇¯ H Xˆs Xs )s=0 = Y, r where ∇¯ is the induced connection for S, ∇¯ H Xˆs is covariant derivative along xs and Xs is the tangent vector of xs , cf., [2]. We have the differential equation ∇¯ H Xˆs ∇¯ H Xˆs Xs + g(∇¯ H Xˆs Xs , ∇¯ H Xˆs Xs )Xs = 0.

(3.2)

By assumption, the curve xs is a circle in M and thus satisfies the differential equation ∇H Xˆs ∇H Xˆs Xs + g(∇H Xˆs Xs ,∇H Xˆs Xs )Xs = 0,

(3.3)

where ∇ is the Cartan covariant derivative for M. Denoting by α the second fundamental form of S in M, hence by operating ∇H Xˆ and using (2.8) we have ∇H Xˆs ∇H Xˆs Xs = ∇H Xˆs (∇¯ H Xˆs Xs ) + ∇H Xˆs (α (H Xˆs ,Xs )) = ∇¯ ˆ ∇¯ ˆ Xs + α (H Xˆs , ∇¯ ˆ Xs ) − A ˆ H Xˆs + ∇¯ ⊥ ˆ α (H Xˆs ,Xs ), H Xs H Xs

α (H Xs ,Xs )

H Xs

H Xs

(3.4)

where AW is the shape operator for a normal vector W and ∇¯ ⊥H Xˆ denotes covariant derivative along xs s relative to the normal connection. Substituting (3.4) into (3.3) and taking into account (3.2), we obtain

α (H Xˆs , ∇¯ H Xˆs Xs ) − Aα (H Xˆs ,Xs ) H Xˆs + ∇¯ ⊥H Xˆ α (H Xˆs ,Xs ) s

+ g(α (H Xˆs ,Xs ), α (H Xˆs ,Xs ))Xs = 0 For the component tangent to S, we get Aα (H Xˆs ,Xs ) H Xˆs = g(α (H Xˆs ,Xs ), α (H Xˆs ,Xs ))Xs .

54

(3.5)


For the component normal to S, we get

α (H Xˆs , ∇¯ H Xˆs Xs ) + ∇¯ ⊥H Xˆ α (H Xˆs ,Xs ) = 0. s

(3.6)

At s = 0, we get noting (∇¯ H Xˆs Xs )s=0 = 1r Y , then we may rewrite (3.6) in the form ˆ ) = −r∇¯ ⊥ ˆ α (H X,X), ˆ α (H X,Y HX

(3.7)

ˆ This equation shows that α (H X,Y ˆ ) = 0 provided X and Y are orthonormal. By where (H Xˆs )s=0 := H X. virtue of Lemma 3.2, we now know that S is umbilical in M, because the assumption of the lemma is valid at each point x ∈ S. Going back to (3.6) once more, we note that α (H Xˆs , ∇¯ H Xˆs Xs ) = 0 since by ¯ = 0 we have g(Xs , ∇¯ ˆ Xs ) = 0. Thus (3.6) gives means of metric compatibility, i.e. ∇g H Xs ∇¯ ⊥H Xˆ α (H Xˆs ,Xs ) = 0. s

(3.8)

By (2) of Lemma 3.2, α (H Xˆs ,Xs ) is equal to the mean curvature vector ηx along the curve xs . At s = 0, (3.8) means that ∇¯ ⊥H Xˆ η = 0. Since x and X ∈ Tx S are arbitrary, we have shown that the mean curvature vector η of S is parallel. Thus S is an extrinsic sphere. Conversely, assume that S is an extrinsic sphere in M. Let xs be a circle in S so that the equation (3.2) is valid. Since S is umbilical, we have

α (H Xˆs ,Xs ) = g(Xs ,Xs )H ηxs = ηxs , Thus (3.4) reduce to ∇H Xˆs ∇H Xˆs Xs = ∇¯ H Xˆs ∇¯ H Xˆs Xs − g(ηxs , ηxs )Xs .

(3.9)

The equation (3.3) is satisfies as a concequence of (3.2) and (3.9). Thus xs is a circle in M and the proof is complete.

References [1] H. Akbar-Zadeh, Initiation to global Finslerian geometry, vol. 68. Elsevier Science, 2006. [2] B. Bidabad, M. Sedaghat, Circle preserving transformations on isotropic Finsler spaces, The 44th Annual Iranian Mathematics Conference, 27-30 August 2013. [3] B. Bidabad, Z. Shen, Circle-preserving transformations in Finsler spaces. Publ. Math. Debrecen, 81 (2012), 435-445. [4] Q. He, W. Yang, W. Zhao, On totally umbilical submanifolds of Finsler spaces, Ann. Polon. Math. 100 (2011), 147-157. [5] J. Li, Umbilical hypersurfaces of Minkowski spaces, Math. Commun. 17 (2012), 63-70. [6] K. Nomizu, K. Yano, On circles and spheres in Riemannian geometry, Mathematische Annalen, 210 (1974), no.2, 163-170.

55

M. Nadjafikhah and P. Kabi-Nejad :: Conservation Laws of the Kupershmidt equation by the Scaling method

Paper No. 1.11

Conservation Laws of the Kupershmidt equation by the Scaling method Mehdi Nadjafikhah1 and Parastoo Kabi-Nejad2 e-mail: m [email protected], Iran University of Science and Technology, Tehran, Iran. 2 e-mail: parastoo [email protected], Iran University of Science and Technology, Tehran, Iran. 1

Abstract In this paper, we derive conservation laws of the fifth order evolutionary integrable Kupershmidt equation that is one of the important models to describe the propagation of the shallow water wave by the scaling method proposed by Hermann et al. (2005). Keywords: Kupershmidt equation, Conservation law, Scaling symmetry

1 Introduction The following fifth order partial differential equation ut = u5x + 5ux u3x + 5u22x − 5u2 u3x − 20uux u2x − 5u3x + 5u4 ux ,

(1)

is known as the Kupershmidt equation [1], [6] that is one of the important models to describe the propagation of the shallow water wave. It is a completely integrable equation and has a tri-Hamiltonian structure with an infinite number of conservation laws and infinitely many symmetries [5]. There are several methods for computing conservation laws as discussed by e.g., Bluman et al. [2], Hereman et al. [3], Naz et al. [7]. One could apply Noether’s theorem, which states that a (variational) symmetry of the PDE corresponds to a conservation law. By contrast, the scaling method [11], uses tools from calculus, the calculus of variations, linear algebra, and differential geometry. Briefly, the method works as follows. A candidate (local) density is assumed to be a linear combination with undetermined coefficients of monomials that are invariant under the scaling symmetry of the PDE. Next, the time derivative of the candidate density is computed and evaluated on the PDE. Subsequently, the variational derivative is applied to get a linear system for the undetermined coefficients. The solution of that system is substituted into the candidate density. Once the density is known, the flux is obtained by applying a homotopy operator to invert a divergence. This paper, is organized as follows. In section 2, we present some definitions and theorems that will be used in the algorithm. In section3, the algorithm is presented and illustrated for the Kupershmidt equation and Additional conservation laws of different ranks of the Kupershmidt equation are given. Finally, some conclutions are drawn in section 4.

56


2 Notations and Definitions In this section, we will provide the background definitions and results that will be used along this paper. Much of it is stated as in [4], [8], [10]. Consider a system of evolution equations ut = P(x, u(M) ),

(2)

where x = (x1 ,· · ·, x p ), and u = (u1 , · · · , uq ) are independent space variables and dependent variables, respectively. A conservation law for (2) is in the form, Dt ρ + DivJ = 0,

on ∆ = 0,

(3)

where ρ is the conserved density and J is the associated flux . In (3), Dt is the total derivative with respect to t and Div is the total Divergence. The algorithm described in section 3 allows one to compute local conservation laws for systems that can be written in the evolutionary form (2). The total derivative operator Dx (in 1 D) acting on f = f (x,t, u(M) (x,t)) of order M is defined as M

j

N 1 ∂f ∂f j Dx f = + ∑ ∑ u(k+1)x j , ∂ x j=1 k=0 ∂u

(4)

kx

j

where M1 is the order of f in component u j and M = max{M11 , · · · , M1N }. The 1D Euler operator for dependent variable u j (x) is defined as j

Lu j (x) f =

M1

∂f

∑ (−Dx)k ∂ u j

k=0

, j = 1, · · · , q.

(5)

kx

The Euler operator allows one to test if differential functions are exact which is a key step in the computation of conservation laws. Let f be a differential function of order M. In 1 D, f is called exact if f is a total derivative, i.e., there exists a differential function F(x, u(M−1) (x)) such that f = Dx F. A differential function f is exact if and only if Lu(x) f = 0. Here, 0 is the vector (0, · · · , 0) which has q components matching the number of components of u. See [9] for more details. Let f be an exact 1 D differential function. The homotopy operator in 1 D is defined as ( ) ∫ 1 N dλ Hu(x) f = Iu j (x) f [λ u] , where u = (u1 , · · · , uq ). (6) ∑ λ 0 j=1 The integrand, I u j (x) f is defined as j

Iu j (x) f =

M1

(

k−1

∑ ∑

k=1

) j uix (−Dx )k−(i+1)

i=0

j

∂f j ∂ ukx

,

(7)

where M1 is the order of f in the dependent variable u j with respect to x. Let f be exact, i.e., Dx F = f for some differential function F(x, u(M−1) (x)). Then, F = D−1 x f = H u(x) f . A proof for the 1 D case in the language of standard calculus is given in [10]. A term or expression f is a divergence if there exists a vector F such that f = DivF. In the 1 D case, f is a total derivative if there exists a function F such that f = Dx F. Note that Dx f is essentially a one-dimensional divergence.

57


3 An Algorithm for Computing a Conservation Law To compute a conservation law, the PDE is assumed to be in the evolutionary form given in (2). The candidate density is constructed by taking a linear combination (with undetermined coefficients) of terms that are invariant under the scaling symmetry of the PDE. The total time derivative of the candidate is computed and evaluated on (2). Hence, all time derivatives from the problem are eliminated. The resulting expression must be exact, so we utilize the Euler operator and Theorem 2.4 to derive the linear system that yields the undetermined coefficients. Substituting these coefficients into the candidate leads to a valid density. Once the density is known the homotopy operator and Theorem 2.6 are used to compute the associated flux J, taking advantage of (2).

3.1 Computing the Scaling Symmetry A PDE has a unique set of Lie-point symmetries which may include translations, rotations, dilations, Galilean boosts, and other symmetries [2]. The application of such symmetries allows one to generate new solutions from known solutions. We will utilize only one type of Lie-point symmetry, namely, the scaling or dilation symmetry, to formulate a candidate density. The Kupershmidt equation (1) is invariant under the scaling symmetry (x,t, u) → (λ −1 x, λ −5t, λ 1 u),

(8)

where λ is an arbitrary scaling parameter. The weight of a variable is defined as the exponent p in the factor λ p that multiplies the variable. For the scaling symmetry x → λ −p x, the weight is denoted W (x) = −p. Total derivatives carry a weight. Indeed, if W (x) = −p, then W (Dx ) = p. The rank of a monomial is the sum of the weights of the variables in the monomial. A differential function is uniform in rank if all monomials in the differential function have the same rank. Hence, we compute the scaling symmetry for the Kupershmidt equation. We assume that the PDE (1) is uniform in rank. Under that assumption, we can form a system of weight-balance equations corresponding to the terms in the PDE. The solution of that system determines the scaling symmetry. The weight-balance equations for the Kupershmidt equation (1) are W (u) +W (Dt ) = W (u) + 5W (Dx ) = 2W (u) + 4W (Dx ) = 3W (u) + 3W (Dx ) = 5W (u) +W (Dx ).

(9)

Solving the linear system gives W (u) = W (Dx ), W (Dt ) = 5W (Dx ). To get (8), set W (Dx ) = 1. The solution to the weight-balance system is then W (u) = 1, W (Dt ) = 5, W (Dx ) = 1,

(10)

3.2 Constructing a Candidate Component Since the Kupershmidt equation (1) has t as evolution variable, we will compute the density ρ of (3) of a fixed rank, for example, R = 6.

58


(a) Construct a list P of differential terms containing all powers of dependent variables and products of dependent variables that have rank 6 or less. In regard to (10), P = {u6 , u5 , u4 , u3 , u2 , u}. (b) Gather all of the terms in P up to rank 6 and put them into a new list, Q. This is done by applying the total derivative operator with respect to the space variable. So, P is replaced by Q = {u6 , u3x , u2 u2x , u4 ux , u22x ,u3 u2x , uux u2x , u2 u3x , ux u3x , uu4x , u5x }.

(11)

(c) In order to construct a nontrivial density with the least number of terms, remove all terms that are divergences or are divergence-equivalent to other terms in Q. So, by applying the Euler operator (5) to each term in (11), we have Lu(x) Q = {6u5 , −6u2x ux , −2uu2x − 2u2 u2x , 0, 2u4x , 6u

2

(12)

u2x + 6uu2x , 3u2x ux , −6u2x ux , 2u4x , 2u4x , 0}.

By Theorem 2.4, u4 ux and u5x are divergences and can be removed from Q. Next, all divergenceequivalent terms will be removed. So, form a linear combination of the terms that remained in (12) with undetermined coefficients pi and set it identically equal to zero, 6p1 u5 − 6p2 u2x ux − 2p3 (uu2x + u2 u2x ) + 2p4 u4x + 6p5 (u2 u2x + uu2x ) +3p6 u2x ux − 6p7 u2x ux + 2p8 u4x + 2p9 u4x = 0.

(13)

Hence, p1 = 0, p6 = 3p2 = 3p7 , p3 = 3p5 , and p4 = −p8 = −p9 . For each divergence-equivalent pair, the terms of highest order are removed from Q in (11). After all divergences and divergenceequivalent terms are cancelled, Q = {u6 , u3x , u2 u2x , u22x }. (d) A candidate density is obtained by forming a linear combination of the remaining terms in Q using undetermined coefficients ci . Thus, the candidate density of rank 6 for the Kupershmidt equation is

ρ = c1 u6 + c2 u3x + c3 u2 u2x + c4 u22x .

(14)

3.3 Evaluating the Undetermined Coefficients Compute the total derivative with respect to t of (14), Dt ρ = (6c1 u5 + 2c3 uu2x )ut + (3c2 u2x + 2c3 u2 ux )uxt + 2c4 u2x u2xt .

(15)

Let E = −Dt ρ after ut , utx and utxx have been replaced using (1). Therefore, by Theorem 2.4, Lu(x) E ≡ 0. Apply the Euler operator to E, gather like terms, and set the result identically equal to zero. Form a linear system for the undetermined coefficients ci . After duplicate equations and common factors have been removed, one gets c2 = −5c1 , c3 = 15c1 , c4 = 3c1 .

(16)

We set c1 = 1 so that the density is normalized on the highest degree term, yielding

ρ = u6 − 5u3x + 15u2 u2x + 3u22x .

59

(17)


3.4 Computing the flux Again, by the continuity equation (2), DivJ = −Dt ρ = E. Therefore, we must compute Div−1 E, where the divergence is the total Derivative with respect to x. After substitution of (16) with c1 = 1 into E and applying the 1 D homotopy operator from Theorem 2.6, we have the following flux: Hu(x) E = 60uux u2x u3x + 30u2x u7 + 75u4 u3x − 120u2x u6 − 3u1 0 − 75u5x

(18)

−30ux u2x u5 − 225u2x u2 u3x − 150uu2x u3x + 330u2x u3 u2x +180ux u4 u3x + 30uu2x u4x − 30ux u2 u5x + 60u2x u2 u4x −60ux u2x u4x − 3u24x + 75u2 u4x − 105u4 u22x − 6u4x u5 +120uu32x − 30u2 u23x + 45u3x u3x + 210u2x u22x + 30ux u23x +15u5x u2x − 60u22x u3x − 6u6x u2x + 6u5x u3x . In the following, additional conservation laws for the kupershmidt equation of rank one and two are obtained,

ρ1 = u, J1 = 5u2 u2x + 5uu2x − u5 − 5ux u2x − u4x , 10 5 ρ2 = u2 , J2 = 10u3 u2x + 5u2 u2x − u6 − 10uux u2x + u3x − 2u4x u + 2ux u3x − u22x . 3 3

(19) (20)

4 Conclusions In this paper, we consider the fifth order evolutionary integrable Kupershmidt equation that admits scaling symmetry and is uniform in rank. so, by the scaling method , the density of rank 6 is constructed and the associated flux is computed by the homotopy operator. Furthermore, additional conservation laws for the kupershmidt equation of rank one and two are given.

References [1] A.H. B ILGE, On the equivalence of linearization and formal symmetries as integrability tests for evolution equations, J. Phys. A 26, No. 24 (1993), 7511-7519. [2] G.W. B LUMAN, A.F. C HEVIAKOV and S.C. A NCO, Applications of Symmetry Methods to Partial Differential Equations, Appl. Math. Sciences, vol. 168, Springer Verlag, New York, 2010. [3] W. H EREMAN, M. C OLAGROSSO, R. S AYERS, A. R INGLER, B. D ECONINCK, M. N IVALA, M.S. H ICKMAN, Continuous and discrete homotopy operators and the computation of conservation laws, In: D. Wang, Z. Zheng, (Eds.), Differential Equations with Symbolic Computation, Birkhauser, Basel (2005), 249-285. [4] W. H EREMAN, Symbolic computation of conservation laws of nonlinear partial differential equations in multi-dimensions, Int. J. Quant. Chem. 106 (2006), 278-299.

60


[5] B.A K UPERSHMIDT, Mathematics of dispersive water waves, Commun. Math. Phys. 99 (1985) 51-73. [6] A.V M IKHAILOV, A.B. S HABAT and V.V S OKOLOV, The symmetry approach to classification of integrable equations, In what is integrability?, Editor, V.E. Zakharov, Springer-Berlin (1991), 115-184. [7] R. NAZ, F.M. M AHOMED and D.P. M ASON, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl. Math. Comput. 205 (2008), 212-230. [8] P.J. O LVER,Application of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York, 1993. [9] L.D. P OOLE, Symbolic computation of conservation laws of nonlinear partial differential equations using homotopy operators, Ph.D. dissertation, Colorado School of Mines, Golden, Colorado, 2009. [10] D. P OOLE and W. H EREMAN, The homotopy operator method for symbolic integration by parts and inversion of divergences with applications, Appl. Anal. 87 (2010) 433-455. [11] D. P OOLE and W. H EREMAN, Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions, Journal of Symbolic Computation, 46 (2011), 1355-1377.

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‫‪Z. Jamal Kashani and B. Bazigaran :: Order‬‬‫‪representability of topological spaces and preorderable‬‬ ‫‪and lower preorderable topologies‬‬

‫‪Paper No. 1.12‬‬ ‫ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﺗﺮﺗﯿﺒﯽ ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ و ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ‬ ‫” داﻧﺸﮕﺎه ﮐﺎﺷﺎن‪ ،‬داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ‪ ،‬ﮔﺮوه رﯾﺎﺿﯽ ﻣﺤﺾ”‬ ‫زﻫﺮه ﺟﻤﺎل ﮐﺎﺷﺎﻧﯽ ‪ 1‬و ﺑﻬﻨﺎم ﺑﺎزﯾﮕﺮان‬

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‫‪ 1‬آدرس ‪١‬‬ ‫‪zohreh [email protected]‬‬ ‫‪ 2‬آدرس‪٢‬‬ ‫‪[email protected]‬‬

‫ﭼﮑﯿﺪه‪ .‬ﻫﺪف اﯾﻦ ﭘﮋوﻫﺶ‪ ،‬ﻣﺮوری اﺳﺖ ﺑﺮ وﯾﮋﮔﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ ) ‪ (CRP‬و ﻧﯿﻢﭘﯿﻮﺳﺘﻪی ) ‪(SRP‬‬ ‫ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ‪ .‬ﺑﺮای اﯾﻦ ﻣﻨﻈﻮر اﺑﺘﺪا ﺑﻪ ﻣﻌﺮﻓﯽ ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ﭘﺮداﺧﺘﻪ‬ ‫و ﺑﺎ اﺳﺘﻔﺎده از آنﻫﺎ‪ ،‬ﺑﺮﺧﯽ از ﺷﺮطﻫﺎی ﻻزم و ﮐﺎﻓﯽ ﺑﺮای آنﮐﻪ ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ در ‪ CRP‬و ‪ SRP‬ﺻﺪق‬ ‫ﮐﻨﺪ را ﺑﯿﺎن ﻣﯽﮐﻨﯿﻢ‪.‬‬

‫‪ .١‬ﭘﯿﺶﮔﻔﺘﺎر‬ ‫ﻣﻄﺎﻟﻌﺎت اﺧﯿﺮ ﺑﺮ روی رواﺑﻂ ﺑﯿﻦ ﺗﺮﺗﯿﺐ و ﺗﻮﭘﻮﻟﻮژی‪ ،‬ﺑﺎ ﭘﯿﺶﺗﺮﺗﯿﺐﻫﺎی ﮐﻠﯽای ﺳﺮوﮐﺎر دارد ﮐﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ‬ ‫ﺗﻮﭘﻮﻟﻮژی داده ﺷﺪه‪ ،‬ﭘﯿﻮﺳﺘﻪ و ﯾﺎ ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ﻫﺴﺘﻨﺪ‪ .‬در ﻫﻤﯿﻦ زﻣﯿﻨﻪ‪ ،‬ﺑﺮرﺳﯽ ﺗﻮﭘﻮﻟﻮژیﻫﺎﯾﯽ ﮐﻪ ﺑﺮ ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ و ﯾﺎ‬ ‫ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ﭘﺎﯾﯿﻨﯽ اﻟﻘﺎ ﺷﺪه ﺗﻮﺳﻂ ﭘﯿﺶ ﺗﺮﺗﯿﺐ ﮐﻠﯽ ﻣﻨﻄﺒﻖ ﻣﯽﺷﻮﻧﺪ‪ ،‬ﻣﻔﯿﺪ ﺧﻮاﻫﺪ ﺑﻮد؛ اﯾﻦ ﺗﻮﭘﻮﻟﻮژیﻫﺎ ﺑﻪ ﺗﺮﺗﯿﺐ‬ ‫ﺗﻮﭘﻮﻟﻮژی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮﻧﺪ‪ .‬ﯾﮑﯽ از راهﻫﺎی ﻣﻄﺎﻟﻌﻪی وﯾﮋﮔﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮی‬ ‫ﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ و ﻧﯿﻢﭘﯿﻮﺳﺘﻪی ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ‪ ،‬ﻣﻄﺎﻟﻌﻪی ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ‬

‫ﻣﯽﺑﺎﺷﺪ ﮐﻪ ﺳﻌﯽ ﺑﺮ آن ﺷﺪه اﺳﺖ ﺗﺎ در اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﻪ ﻣﻌﺮﻓﯽ اﯾﻦ ﺗﻮﭘﻮﻟﻮژیﻫﺎ ﭘﺮداﺧﺘﻪ‪ ،‬ﭘﺎﯾﻪای ﺑﺮای آنﻫﺎ ﻣﻌﺮﻓﯽ ﮐﺮده‬ ‫و ﺳﭙﺲ ﺑﻪ ﺑﺮرﺳﯽ وﯾﮋﮔﯽﻫﺎی ﺑﯿﺎن ﺷﺪه ﺑﭙﺮدازﯾﻢ‪ .‬اﺑﺘﺪا ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ را ﺑﻪ ﻋﻨﻮان اﺑﺰاری ﻣﻬﻢ ﻣﻌﺮﻓﯽ ﻣﯽﮐﻨﯿﻢ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ (X, -) .١.١‬را ﻣﺠﻤﻮﻋﻪی ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ‪ .‬ﺧﺎﻧﻮادهی ﻫﻤﻪی ﻣﺠﻤﻮﻋﻪﻫﺎی ﺑﻪ ﺷﮑﻞ‬ ‫}‪ L(x) = {a ∈ X : a ≺ x‬و }‪ G(x) = {a ∈ X : x ≺ a‬ﮐﻪ ‪ ،x ∈ X‬ﯾﮏ زﯾﺮﭘﺎﯾﻪ ﺑﺮای ﺗﻮﭘﻮﻟﻮژی‬ ‫‪ T‬روی ‪ X‬اﺳﺖ‪ T- .‬را ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ روی ‪ X‬ﻧﺎﻣﻨﺪ و ﺟﻔﺖ ) ‪ (X, T-‬ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﻧﺎﻣﯿﺪه‬‫ﻣﯽﺷﻮد‪.‬‬ ‫ﻣﺸﺎﻫﺪه ﮐﻨﯿﺪ ﮐﻪ ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ‪ T-‬اﺟﺘﻤﺎع دو ﺗﻮﭘﻮﻟﻮژی اﺳﺖ ﺑﻪ ﻧﺎمﻫﺎی ﺗﻮﭘﻮﻟﻮژی ﭘﺎﯾﯿﻨﯽ ‪ ،T-l‬ﮐﻪ ﺧﺎﻧﻮادهی‬

‫ﻫﻤﻪی ﻣﺠﻤﻮﻋﻪﻫﺎی ﺑﻪ ﺷﮑﻞ }‪ G(x) = {a ∈ X : x ≺ a‬زﯾﺮﭘﺎﯾﻪی آن اﺳﺖ‪ ،‬و ﺗﻮﭘﻮﻟﻮژی ﺑﺎﻻﯾﯽ ‪ ،T-u‬ﮐﻪ‬ ‫ﺧﺎﻧﻮادهی ﻫﻤﻪی ﻣﺠﻤﻮﻋﻪﻫﺎی ﺑﻪ ﺷﮑﻞ }‪ L(x) = {a ∈ X : a ≺ x‬زﯾﺮﭘﺎﯾﻪی آن ﻣﯽﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .٢.١‬اﻟﻒ( اﮔﺮ )‪ (X, -‬ﯾﮏ ﻣﺠﻤﻮﻋﻪی ﭘﯿﺶﺗﺮﺗﯿﺐ و ‪ T‬ﯾﮏ ﺗﻮﭘﻮﻟﻮژی روی ‪ X‬ﺑﺎﺷﺪ‪ ،‬آنﮔﺎه ﭘﯿﺶﺗﺮﺗﯿﺐ‬ ‫ روی ‪ X‬را‪-T ،‬ﭘﯿﻮﺳﺘﻪ روی ‪ X‬ﮔﻮﯾﻨﺪ‪ ،‬ﻫﺮﮔﺎه ﺑﺮای ﻫﺮ ‪ x ∈ X‬ﻣﺠﻤﻮﻋﻪﻫﺎی‬‫}‪ {a ∈ X : x - a‬و }‪ {b ∈ X : b - x‬در ‪-T ، X‬ﺑﺴﺘﻪ ﺑﺎﺷﻨﺪ‪.‬‬ ‫ب( ﭘﯿﺶﺗﺮﺗﯿﺐ ‪ -‬را ‪-T‬ﻧﯿﻢﭘﯿﻮﺳﺘﻪی ﭘﺎﯾﯿﻨﯽ روی ‪ X‬ﮔﻮﯾﻨﺪ‪ ،‬ﻫﺮﮔﺎه ﺑﺮای ﻫﺮ ‪ x ∈ X‬ﻣﺠﻤﻮﻋﻪی‬ ‫}‪ {a ∈ X : a - x‬در ‪-T ، X‬ﺑﺴﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫پ( ﭘﯿﺶﺗﺮﺗﯿﺐ ‪ -‬را ‪ -T‬ﻧﯿﻢﭘﯿﻮﺳﺘﻪی ﺑﺎﻻﯾﯽ روی ‪ X‬ﮔﻮﯾﻨﺪ‪ ،‬ﻫﺮﮔﺎه ﺑﺮای ﻫﺮ ‪ x ∈ X‬ﻣﺠﻤﻮﻋﻪی‬ ‫}‪ {a ∈ X : x - a‬در ‪-T ، X‬ﺑﺴﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .٣.١‬اﮔﺮ )‪ (X, -‬ﯾﮏ ﻣﺠﻤﻮﻋﻪی ﭘﯿﺶﺗﺮﺗﯿﺐ ﺑﺎﺷﺪ‪ ،‬ﺗﺎﺑﻊ ‪ u : X → R‬ﮔﻔﺘﻪ ﻣﯽﺷﻮد‪:‬‬ ‫آ( ﺻﻌﻮدی اﮔﺮ ﺑﺮای ﻫﺮ ‪ x, y ∈ X‬ﮐﻪ ‪ x - y‬ﻧﺘﯿﺠﻪ ﺷﻮد )‪.u(x) 6 u(y‬‬ ‫ب( ﺣﺎﻓﻆ ﺗﺮﺗﯿﺐ )اﯾﺰوﺗﻮن( اﮔﺮ ‪ u‬ﺻﻌﻮدی ﺑﺎﺷﺪ و )‪.x ≺ y ⇒ u(x) < u(y‬‬

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‫ﯾﮏ ﺗﺎﺑﻊ ﺣﺎﻓﻆ ﺗﺮﺗﯿﺐ ﻫﻤﭽﻨﯿﻦ ﯾﮏ ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .۴.١‬اﮔﺮ ﺑﻪ ﻣﺠﻤﻮﻋﻪی ﻧﺎﺗﻬﯽ ‪ X‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬داده ﺷﻮد‪ ،‬آنﮔﺎه ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ‪ -‬روی ‪ X‬را ﻧﻤﺎﯾﺶﭘﺬﯾﺮ‬ ‫ﭘﯿﻮﺳﺘﻪ ﮔﻮﯾﻨﺪ‪ ،‬ﻫﺮﮔﺎه ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻮﭘﻮﻟﻮژی ‪ T‬روی ‪ X‬و ﺗﻮﭘﻮﻟﻮژی ﻣﻌﻤﻮل‬ ‫روی ﺧﻂ ﺣﻘﯿﻘﯽ ‪ R‬ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .۵.١‬ﻣﺠﻤﻮﻋﻪی ﻧﺎﺗﻬﯽ ‪ X‬را ﺑﺎ ﺗﻮﭘﻮﻟﻮژی داده ﺷﺪهی ‪ T‬در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ‪ .‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬روی ‪ X‬وﯾﮋﮔﯽ‬ ‫ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﭘﯿﻮﺳﺘﻪ ‪ CRP‬دارد‪ ،‬اﮔﺮ ﻫﺮ ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ﭘﯿﻮﺳﺘﻪ ‪ -‬ﺗﻌﺮﯾﻒ ﺷﺪه روی ‪ ،X‬دارای ﻧﻤﺎﯾﺸﯽ ﺑﻪ‬ ‫ﺻﻮرت ﯾﮏ ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .۶.١‬ﻣﺠﻤﻮﻋﻪی ﻧﺎﺗﻬﯽ ‪ X‬را ﺑﺎ ﺗﻮﭘﻮﻟﻮژی داده ﺷﺪهی ‪ T‬در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ ‪ .‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬روی ‪ X‬وﯾﮋﮔﯽ‬ ‫ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ‪ SRP‬دارد‪ ،‬اﮔﺮ ﻫﺮ ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ‪ -‬ﺗﻌﺮﯾﻒ ﺷﺪه روی ‪ ،X‬دارای ﻧﻤﺎﯾﺸﯽ‬ ‫ﺑﻪ ﺻﻮرت ﯾﮏ ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .٧.١‬ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ) ‪ (X, T‬را در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ‪ .‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬را ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ )ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ‬ ‫ﭘﺎﯾﯿﻨﯽ( ﻧﺎﻣﻨﺪ‪ ،‬ﻫﺮﮔﺎه ﭘﯿﺶﺗﺮﺗﯿﺐ ‪ -‬ﺑﺮ ‪ X‬وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ﺗﻮﭘﻮﻟﻮژی ‪ T‬ﺑﺮ ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ‪) T-‬ﺗﻮﭘﻮﻟﻮژی‬ ‫ﺗﺮﺗﯿﺒﯽ ﭘﺎﯾﯿﻨﯽ ‪ (T-l‬ﻣﻨﻄﺒﻖ ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .٨.١‬ﻓﺮض ﮐﻨﯿﺪ ) ‪ (X, T‬ﻣﺠﻤﻮﻋﻪای ﭘﯿﺶﺗﺮﺗﯿﺐ ﺑﺎﺷﺪ‪ .‬ﮔﻮﯾﯿﻢ ﭘﯿﺶﺗﺮﺗﯿﺐ ‪ -‬ﺟﺪاﯾﯽﭘﺬﯾﺮ ﺗﺮﺗﯿﺒﯽ از ﻧﻈﺮ‬ ‫دﺑﺮﯾﻮ ‪ ١‬اﺳﺖ ﻫﺮﮔﺎه زﯾﺮﻣﺠﻤﻮﻋﻪی ﺷﻤﺎرای ‪ Z‬از ‪ X‬وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ ﺑﻪﻃﻮری ﮐﻪ ﻫﺮﮔﺎه ‪ z ∈ Z، x ≺ y‬وﺟﻮد‬ ‫داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ ‪.x - z - y‬‬ ‫ﻗﻀﯿﻪ ‪ .٩.١‬ﻓﺮض ﮐﻨﯿﺪ )‪ (X, -‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎﺷﺪ‪[١].‬‬ ‫‪ (١‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬ﭘﯿﺶﺗﺮﺗﯿﺐ ﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﯾﮏ ﭘﺎﯾﻪ ﻣﺜﻞ‬ ‫}‪ B = {Oα ⊆ X : α ∈ A‬داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ در دو ﺷﺮط زﯾﺮ ﺻﺪق ﻣﯽﮐﻨﺪ )ﮐﻪ ‪ A‬ﻣﺠﻤﻮﻋﻪی‬ ‫اﻧﺪﯾﺲ اﺳﺖ(‪:‬‬ ‫آ( ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ‪ Oα ⊆ Oβ‬ﯾﺎ ‪Oβ ⊆ Oα‬؛‬ ‫∩‬ ‫ب( ﺑﺮای ﻫﺮ ‪ α ∈ A‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ∅ ≠ ) ‪(Oγ \ Oα‬‬ ‫‪γ∈A‬‬ ‫‪Oα (Oγ‬‬

‫‪ (٢‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬ﭘﯿﺶﺗﺮﺗﯿﺐ ﭘﺬﯾﺮ اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﯾﮏ زﯾﺮ ﭘﺎﯾﻪ ﻣﺜﻞ‬ ‫∪‬ ‫}‪S = {Oα ⊆ X : α ∈ A} {Px : x ∈ X‬‬ ‫داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﻪ در ﺳﻪ ﺷﺮط زﯾﺮ ﺻﺪق ﻣﯽﮐﻨﺪ‪:‬‬ ‫آ( ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ‪ Oα ⊆ Oβ‬ﯾﺎ ‪Oβ ⊆ Oα‬؛‬ ‫∩‬ ‫؛‬ ‫ب( ﺑﺮای ﻫﺮ ‪ α ∈ A‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ∅ ≠ ) ‪(Oγ \ Oα‬‬ ‫ج( ﺑﺮای ﻫﺮ ‪ x ∈ X‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ) ‪(X \ Oα‬‬

‫∪‬

‫‪γ∈A‬‬ ‫‪Oα (Oγ‬‬

‫= ‪.Px‬‬

‫‪α∈A‬‬ ‫‪x∈Oα‬‬

‫‪ .٢‬ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﭘﯿﻮﺳﺘﻪ و ﻧﯿﻢﭘﯿﻮﺳﺘﻪ‬ ‫اﯾﻦ ﺑﺨﺶ را ﺑﺎ ﻗﻀﯿﻪای ﺷﺮوع ﻣﯽﮐﻨﯿﻢ ﮐﻪ ﺑﺎ اﺳﺘﻔﺎده از ﺗﻌﺮﯾﻒ ﭘﺎﯾﻪی ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ‬ ‫ﭘﺎﯾﯿﻨﯽ‪ ،‬ﺷﺮط ﻻزم و ﮐﺎﻓﯽ ﺑﺮای اﯾﻦﮐﻪ ﯾﮏ ﺗﻮﭘﻮﻟﻮژی در ‪ CRP‬و ‪ SRP‬ﺻﺪق ﮐﻨﺪ را ﺑﯿﺎن ﻣﯽﮐﻨﺪ‪ .‬در اداﻣﻪ‬ ‫ﺧﺎﻧﻮادهﻫﺎی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ و ﻧﯿﻢﭘﯿﻮﺳﺘﻪ را ﻣﻌﺮﻓﯽ ﮐﺮده ﺗﺎ ﺷﺮاﯾﻂ را ﺑﺮای ﺑﯿﺎن ﻗﻀﯿﻪای دﯾﮕﺮ‪ ،‬ﮐﻪ ﺷﺮط ﻻزم‬ ‫و ﮐﺎﻓﯽ ﺑﺮای ﺻﺪق ﮐﺮدن ﺗﻮﭘﻮﻟﻮژی ﯾﮏ ﻓﻀﺎ در ‪ CRP‬و ‪ SRP‬را ﻣﻄﺮح ﻣﯽﮐﻨﺪ‪ ،‬ﻓﺮاﻫﻢ ﮐﻨﯿﻢ‪.‬‬

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‫ﻗﻀﯿﻪ ‪ .١.٢‬ﻓﺮض ﮐﻨﯿﺪ ) ‪ (X, T‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎﺷﺪ‪ .‬ﺗﻮﭘﻮﻟﻮژی ‪ T‬در ‪ (SRP ) CRP‬ﺻﺪق ﻣﯽﮐﻨﺪ‬ ‫اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ زﯾﺮﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ) ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ( ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم ﺑﺎﺷﺪ‪[١] .‬‬ ‫ﺑﺎ اﺳﺘﻔﺎده از اﯾﻦ ﻗﻀﯿﻪ ﻣﯽﺗﻮاﻧﯿﻢ راﺑﻄﻪی ﺑﯿﻦ ‪ CRP‬و ‪ SRP‬را ﭘﯿﺪا ﮐﻨﯿﻢ‪ ،‬اﻟﺒﺘﻪ ﺑﺮای ﺑﯿﺎن اﯾﻦ راﺑﻄﻪ ﺑﻪ ﯾﮏ‬ ‫ﻟﻢ و ﻗﻀﯿﻪ ﻧﯿﺎز دارﯾﻢ‪.‬‬ ‫ﻟﻢ ‪ .٢.٢‬ﻓﺮض ﮐﻨﯿﺪ ‪ X‬ﯾﮏ ﻣﺠﻤﻮﻋﻪی ﻧﺎﺗﻬﯽ ﺑﺎ ﯾﮏ ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ‪ -‬و ‪ T-‬ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ روی ‪ X‬ﺑﺎﺷﺪ‪.‬‬ ‫در اﯾﻦ ﺻﻮرت ﺷﺮاﯾﻂ زﯾﺮ ﻣﻌﺎدﻟﻨﺪ‪ [١] :‬و ]‪[٢‬‬ ‫آ(‬ ‫ب(‬ ‫پ(‬ ‫ت(‬

‫ﯾﮏ ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ از )‪ (X, -‬ﺑﻪ ﺧﻂ ﺣﻘﯿﻘﯽ ‪ R‬ﺑﺎ ﺗﺮﺗﯿﺐ ﻣﻌﻤﻮﻟﯽ ‪ 6‬وﺟﻮد دارد‪.‬‬ ‫ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ‪ ) -‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ‪ T-‬روی ‪ X‬و ﺗﻮﭘﻮﻟﻮژی ﻣﻌﻤﻮل روی ‪ ( R‬ﺑﻪ واﺳﻄﻪی‬ ‫ﯾﮏ ﺗﮑﺮﯾﺨﺘﯽ ﺗﺮﺗﯿﺒﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮ ﭘﯿﻮﺳﺘﻪ اﺳﺖ‪.‬‬ ‫ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ‪ T-‬ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم اﺳﺖ‪.‬‬ ‫ﻣﺠﻤﻮﻋﻪی ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ )‪ (X, -‬ﺟﺪاﯾﯽﭘﺬﯾﺮ ﺗﺮﺗﯿﺒﯽ دﺑﺮﯾﻮ اﺳﺖ‪.‬‬

‫ﻗﻀﯿﻪ ‪ X .٣.٢‬را ﻣﺠﻤﻮﻋﻪای ﻧﺎﺗﻬﯽ ﺑﺎ ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ‪ -‬در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ‪ .‬در اﯾﻦ ﺻﻮرت ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ‪T-‬‬ ‫ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم اﺳﺖ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﺗﻮﭘﻮﻟﻮژی ﺗﺮﺗﯿﺒﯽ ﭘﺎﯾﯿﻨﯽ ‪ T-l‬ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم ﺑﺎﺷﺪ‪[١] .‬‬ ‫ﮔﺰاره ‪ .۴.٢‬اﮔﺮ ﺗﻮﭘﻮﻟﻮژی ‪ T‬در ‪ SRP‬ﺻﺪق ﮐﻨﺪ‪ ،‬در ‪ CRP‬ﻧﯿﺰ ﺻﺪق ﺧﻮاﻫﺪ ﮐﺮد‪.‬‬ ‫ﺑﺮﻫﺎن‪ .‬ﻓﺮض ﻣﯽﮐﻨﯿﻢ ﺗﻮﭘﻮﻟﻮژی ‪ T‬در ‪ SRP‬ﺻﺪق ﮐﻨﺪ‪ ،‬در اﯾﻦ ﺻﻮرت ﻃﺒﻖ ﻗﻀﯿﻪی ‪ ،١.٢‬زﯾﺮﺗﻮﭘﻮﻟﻮژیﻫﺎی‬ ‫ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم ﻫﺴﺘﻨﺪ‪ .‬ﺑﻪ وﺿﻮح ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﺗﺮﺗﯿﺒﯽ ﭘﺎﯾﯿﻨﯽ ‪ ،T-l‬ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ‬ ‫ﭘﺎﯾﯿﻨﯽ و در ﻧﺘﯿﺠﻪ ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم ﻫﺴﺘﻨﺪ ﮐﻪ اﯾﻦ ﻃﺒﻖ ﻗﻀﯿﻪی ‪ ٣.٢‬ﻣﻌﺎدل اﺳﺖ ﺑﺎ اﯾﻦﮐﻪ ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﺗﺮﺗﯿﺒﯽ‬ ‫‪ ،T‬ﺷﻤﺎرشﭘﺬﯾﺮ ﻧﻮع دوم ﺑﺎﺷﻨﺪ‪ .‬در ﻧﻬﺎﯾﺖ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻟﻢ ‪ ٢.٢‬ﺑﻪ اﯾﻦ ﺣﻘﯿﻘﺖ ﻣﯽرﺳﯿﻢ ﮐﻪ ﺗﻮﭘﻮﻟﻮژی ﻣﻮرد ﻧﻈﺮ در‬‫‬ ‫‪ CRP‬ﺻﺪق ﻣﯽﮐﻨﺪ‪.‬‬ ‫ﺗﻮﺟﻪ ‪ .۵.٢‬در ]‪ [٣‬ﻧﺸﺎن داده ﻣﯽﺷﻮد ﮐﻪ در ﺣﺎﻟﺖ ﮐﻠﯽ ‪ SRP ،CRP‬را ﻧﺘﯿﺠﻪ ﻧﻤﯽدﻫﺪ اﻣﺎ در ﻓﻀﺎﻫﺎی ﻣﺘﺮﯾﮏ‬ ‫اﯾﻦ دو وﯾﮋﮔﯽ ﺑﺎ ﯾﮑﺪﯾﮕﺮ ﻣﻌﺎدﻟﻨﺪ‪.‬‬ ‫ﺑﺎ اﺳﺘﻔﺎده از وﯾﮋﮔﯽ ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ و ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ﮐﻪ در ﻗﻀﯿﻪی ‪ ٩.١‬ﺑﯿﺎن ﺷﺪ‪ ،‬ﻣﯽﺗﻮاﻧﯿﻢ‬

‫ﺧﺼﻮﺻﯿﺎت ﺑﯿﺸﺘﺮی از ﺗﻮﭘﻮﻟﻮژیﻫﺎﯾﯽ ﮐﻪ در ‪ CRP‬و ‪ SRP‬ﺻﺪق ﻣﯽﮐﻨﻨﺪ را ﺑﺪﺳﺖ آورﯾﻢ‪ .‬ﺑﺮای اﯾﻦ ﻣﻨﻈﻮر اﺑﺘﺪا‬ ‫ﭼﻨﺪ ﺗﻌﺮﯾﻒ اراﺋﻪ ﻣﯽدﻫﯿﻢ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .۶.٢‬ﻓﺮض ﮐﻨﯿﺪ ) ‪ (X, T‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ و ﻣﺠﻤﻮﻋﻪی ‪ S‬ﮐﻪ ﺑﻪﺻﻮرت زﯾﺮ ﺗﻌﺮﯾﻒ ﻣﯽﺷﻮد ﺧﺎﻧﻮادهای‬ ‫از زﯾﺮﻓﻀﺎﻫﺎی ‪ X‬ﺑﺎﺷﺪ )‪ A‬ﻣﺠﻤﻮﻋﻪای ﻧﺎﺗﻬﯽ از اﻧﺪﯾﺲﻫﺎ را ﻣﺸﺨﺺ ﻣﯽﮐﻨﺪ(‪.‬‬ ‫∪‬ ‫}‪S = {Oα ⊆ X : α ∈ A} {Px : x ∈ X‬‬ ‫ﺧﺎﻧﻮادهی ‪ S‬ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد اﮔﺮ در ﺷﺮاﯾﻂ زﯾﺮ ﺻﺪق ﮐﻨﺪ‪:‬‬ ‫آ( ﻋﻨﺎﺻﺮ ‪ Oα‬و ‪ (α ∈ A, x ∈ X) Px‬ﻫﻤﮕﯽ ‪-T‬ﺑﺎز ﺑﺎﺷﻨﺪ؛‬ ‫ب( ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬اﯾﻦ ﺑﺮﻗﺮار ﺑﺎﺷﺪ ﮐﻪ ‪ Oα ⊆ Oβ‬ﯾﺎ ‪Oβ ⊆∩Oα‬؛‬ ‫؛‬ ‫پ( ﺑﺮای ﻫﺮ ‪ α ∈ A‬اﯾﻦ ﺑﺮﻗﺮار ﺑﺎﺷﺪ ﮐﻪ ∅ ≠ ) ‪(Oγ \ Oα‬‬ ‫‪γ∈A‬‬ ‫∪‬ ‫‪Oα (Oγ‬‬ ‫= ‪.Px‬‬ ‫ت( ﺑﺮای ﻫﺮ ‪ x ∈ X‬داﺷﺘﻪ ﺑﺎﺷﯿﻢ ) ‪(X \ Oα‬‬ ‫‪α∈A‬‬ ‫‪x∈Oα‬‬

‫ﺗﻌﺮﯾﻒ ‪ .٧.٢‬ﻓﺮض ﮐﻨﯿﺪ ) ‪ (X, T‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ و }‪ F = {Oα ⊆ X : α ∈ A‬ﺧﺎﻧﻮادهای از‬ ‫زﯾﺮﻣﺠﻤﻮﻋﻪﻫﺎی ‪ X‬ﺑﺎﺷﺪ )‪ A‬ﻣﺠﻤﻮﻋﻪای ﻧﺎﺗﻬﯽ از اﻧﺪﯾﺲﻫﺎ را ﻣﺸﺨﺺ ﻣﯽﮐﻨﺪ(‪ .‬ﺧﺎﻧﻮادهی ‪ F‬ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ‬ ‫ﻧﯿﻢ ﭘﯿﻮﺳﺘﻪ ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد‪ ،‬اﮔﺮ در ﺷﺮاﯾﻂ زﯾﺮ ﺻﺪق ﮐﻨﺪ‪:‬‬ ‫آ( ﻫﺮ ‪-T ، Oα‬ﺑﺎز ﺑﺎﺷﻨﺪ؛‬ ‫ب( ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬اﯾﻦ ﺑﺮﻗﺮار ﺑﺎﺷﺪ ﮐﻪ ‪ Oα ⊆ Oβ‬ﯾﺎ ‪Oβ ⊆ Oα‬؛‬ ‫‪64‬‬


‫∩‬

‫پ( ﺑﺮای ﻫﺮ ‪ α ∈ A‬اﯾﻦ ﺑﺮﻗﺮار ﺑﺎﺷﺪ ﮐﻪ ∅ ≠ ) ‪(Oγ \ Oα‬‬

‫‪.‬‬

‫‪γ∈A‬‬ ‫‪Oα (Oγ‬‬

‫در ﺣﻘﯿﻘﺖ ﯾﮏ راﺑﻄﻪی دوﺳﻮﯾﯽ ﺑﯿﻦ ”ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ )ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ(” و ”ﺧﺎﻧﻮادهی‬ ‫ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ )ﻧﯿﻢﭘﯿﻮﺳﺘﻪ(” وﺟﻮد دارد‪ .‬ﺑﺮای اﺛﺒﺎت اﯾﻦ ﻣﻮﺿﻮع ﺑﻪ ﺻﻮرت زﯾﺮ ﻋﻤﻞ ﻣﯽﮐﻨﯿﻢ‪:‬‬ ‫اﮔﺮ ‪ S‬ﯾﮏ ﺧﺎﻧﻮادهی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ ﺑﺎﺷﺪ‪ ،‬راﺑﻄﻪی دوﺗﺎﯾﯽ زﯾﺮ را ﮐﻪ ﺑﻪ راﺣﺘﯽ اﺛﺒﺎت ﻣﯽﺷﻮد ﯾﮏ ﭘﯿﺶﺗﺮﺗﯿﺐ‬ ‫ﮐﻠﯽ اﺳﺖ را در ﻧﻈﺮ ﻣﯽﮔﯿﺮﯾﻢ‪:‬‬ ‫)‪x -S y ⇔ (x ∈ Oα ⇒ y ∈ Oα ) ∀α ∈ A (x, y ∈ X‬‬ ‫∩‬ ‫∪‬ ‫∪‬ ‫\ ‪ ،L(x) = X‬ﮐﻪ‬ ‫= ‪Oα‬‬ ‫= )‪ G(x‬و ‪(X \ Oα ) = Px‬‬ ‫ﺑﺮای ﻫﺮ ‪ x ∈ X‬دارﯾﻢ ‪Oα‬‬ ‫‪α∈A‬‬ ‫‪x̸∈Oα‬‬

‫‪α∈A‬‬ ‫‪x̸∈Oα‬‬

‫‪α∈A‬‬ ‫‪x∈Oα‬‬

‫ﻧﺘﯿﺠﻪ ﻣﯽﺷﻮد ‪ G(x), L(x) ∈∩T‬و در ﻧﺘﯿﺠﻪ ‪.T-S ⊆ T‬ﻋﻼوه ﺑﺮ اﯾﻦ ﭼﻮن ‪ α ∈ A‬وﺟﻮد دارد ﮐﻪ ﺑﺮای ﻫﺮ‬ ‫∈ ‪ x‬دارﯾﻢ )‪ Oα = G(x‬و از ﻃﺮﻓﯽ )‪ Px = L(x‬ﻟﺬا ﻧﺘﯿﺠﻪ ﻣﯽﮔﯿﺮﯾﻢ ‪ T ⊆ T-S‬وﻟﺬا‬ ‫) ‪(Oγ \ Oα‬‬ ‫‪γ∈A‬‬ ‫‪Oα (Oγ‬‬

‫‪ T‬و ‪ T-S‬ﺑﺮ ﻫﻢ ﻣﻨﻄﺒﻖاﻧﺪ و ‪ T‬ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﻣﯽﺷﻮد‪ .‬ﺣﺎل اﮔﺮ ‪ F‬ﯾﮏ ﺧﺎﻧﻮادهی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﻧﯿﻢﭘﯿﻮﺳﺘﻪ‬ ‫ﺑﺎﺷﺪ‪ ،‬راﺑﻄﻪی دوﺗﺎﯾﯽ زﯾﺮ را ﮐﻪ ﺑﻪ راﺣﺘﯽ اﺛﺒﺎت ﻣﯽﺷﻮد ﯾﮏ ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ اﺳﺖ را در ﻧﻈﺮ ﻣﯽﮔﯿﺮﯾﻢ‪:‬‬ ‫) ‪x -F y ⇔ (x ∈ Oα ⇒ y ∈ Oα‬‬

‫)‪∀α ∈ A (x, y ∈ X‬‬ ‫∪‬ ‫= )‪ G(x‬ﮐﻪ ﻧﺘﯿﺠﻪ ﻣﯽﺷﻮد ‪ G(x) ∈ T‬و در ﻧﺘﯿﺠﻪ ‪⊆ T‬‬ ‫دوﺑﺎره ﺑﺮای ﻫﺮ ‪ x ∈ X‬دارﯾﻢ ‪Oα‬‬ ‫‪α∈A‬‬ ‫‪x̸∈Oα‬‬

‫ﻋﻼوه ﺑﺮ اﯾﻦ ﭼﻮن ‪ α ∈ A‬وﺟﻮد دارد ﮐﻪ ﺑﺮای ﻫﺮ ) ‪(Oγ \ Oα‬‬

‫∩‬

‫‪.T-F‬‬

‫∈ ‪ x‬دارﯾﻢ )‪ Oα = G(x‬ﻧﺘﯿﺠﻪ‬

‫‪γ∈A‬‬ ‫‪Oα (Oγ‬‬

‫ﻣﯽﮔﯿﺮﯾﻢ ‪ T ⊆ T-F‬وﻟﺬا ‪ T‬و ‪ T-F‬ﺑﺮ ﻫﻢ ﻣﻨﻄﺒﻖاﻧﺪ و ‪ T‬ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ ﻣﯽﺷﻮد‪ .‬ﺑﺮﻋﮑﺲ‪ ،‬اﮔﺮ ‪ T‬ﯾﮏ‬ ‫ﺗﻮﭘﻮﻟﻮژی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ )ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ( ﺑﺎﺷﺪ‪ ،‬ﻃﺒﻖ ﻗﻀﯿﻪی ‪ ٩.١‬زﯾﺮﭘﺎﯾﻪای )ﭘﺎﯾﻪای( دارد ﮐﻪ در ﺷﺮاﯾﻂ‬ ‫ﮔﻔﺘﻪ ﺷﺪه در آن ﻗﻀﯿﻪ ﺻﺪق ﻣﯽﮐﻨﺪ و ﻫﻤﺎنﻃﻮر ﮐﻪ دﯾﺪه ﻣﯽﺷﻮد اﯾﻦ ﺷﺮاﯾﻂ ﻫﻤﺎن ﺷﺮاﯾﻂ ﺧﺎﻧﻮادهی ﭘﯿﺶﺗﺮﺗﯿﺒﯽ‬ ‫ﭘﯿﻮﺳﺘﻪ )ﻧﯿﻢﭘﯿﻮﺳﺘﻪ( اﺳﺖ‪ .‬ﺑﻨﺎﺑﺮاﯾﻦ‪ ،‬ﻧﺘﯿﺠﻪ ﻣﯽﮔﯿﺮﯾﻢ ﮐﻪ در ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ) ‪ ،(X, T‬راﺑﻄﻪای دو ﺳﻮﯾﯽ ﺑﯿﻦ‬ ‫”ﺗﻮﭘﻮﻟﻮژیﻫﺎی ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ )ﭘﯿﺶﺗﺮﺗﯿﺐﭘﺬﯾﺮ ﭘﺎﯾﯿﻨﯽ(” و ”ﺧﺎﻧﻮادهی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ )ﻧﯿﻢﭘﯿﻮﺳﺘﻪ(” وﺟﻮد‬ ‫دارد‪.‬‬ ‫اﯾﻦ ﺣﻘﯿﻘﺖ ﺑﻪ ﻣﺎ اﯾﻦ اﺟﺎزه را ﻣﯽدﻫﺪ ﮐﻪ ﺻﻔﺎت ‪ CRP‬و ‪ SRP‬را روی ﻓﻀﺎﻫﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ) ‪ (X, T‬را ﺑﯿﺎن‬ ‫ﮐﻨﯿﻢ و ﺑﺎ ﺧﺎﻧﻮادهﻫﺎی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪ و ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ﮐﺎر ﮐﻨﯿﻢ‪ .‬ﺑﺮای اﯾﻦ ﻣﻨﻈﻮر ﺗﻌﺮﯾﻔﯽ دﯾﮕﺮ اراﺋﻪ ﻣﯽدﻫﯿﻢ‪.‬‬ ‫ﺗﻌﺮﯾﻒ ‪ .٨.٢‬ﻓﺮض ﮐﻨﯿﺪ ) ‪ (X, T‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎﺷﺪ و }‪ T = {Tα ⊆ X : α ∈ A‬ﺧﺎﻧﻮادهای‬ ‫زﯾﺮﻣﺠﻤﻮﻋﻪﻫﺎی ‪ X‬ﺑﺎﺷﺪ‪ ،‬ﮐﻪ ‪ A‬ﻣﺠﻤﻮﻋﻪای از اﻧﺪﯾﺲﻫﺎ اﺳﺖ‪ .‬ﻋﻨﺼﺮ ‪ x ∈ X‬را در ﻧﻈﺮ ﺑﮕﯿﺮﯾﺪ‪ .‬ﻣﺠﻤﻮﻋﻪی‬ ‫از ∪‬ ‫⋆‬ ‫= ‪ Tx‬را ﻣﺠﻤﻮﻋﻪی ﻏﯿﺎب ‪ x‬ﻧﺴﺒﺖ ﺑﻪ ﺧﺎﻧﻮادهی ‪ T‬ﮔﻮﯾﻨﺪ‪) .‬ﻣﺸﺎﻫﺪه ﮐﻨﯿﺪ ﮐﻪ وﻗﺘﯽ ‪ x‬در ﻫﻤﻪی ‪Tα‬ﻫﺎ‬ ‫‪Tα‬‬ ‫‪α∈A‬‬ ‫‪x̸∈Tα‬‬

‫ﺑﺎﺷﺪ‪ Tx⋆ ،‬ﺗﻬﯽ ﻣﯽﺷﻮد‪( .‬‬ ‫ﻗﻀﯿﻪ ‪ .٩.٢‬ﻓﺮض ﮐﻨﯿﺪ ﮐﻪ ) ‪ (X, T‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎﺷﺪ‪.‬‬ ‫آ( ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ) ‪ (X, T‬در وﯾﮋﮔﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﭘﯿﻮﺳﺘﻪ ‪ CRP‬ﺻﺪق ﻣﯽﮐﻨﺪ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﺑﺮای ﻫﺮ‬ ‫ﺧﺎﻧﻮادهی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﭘﯿﻮﺳﺘﻪی‬ ‫∪‬ ‫}‪S = {Oα ⊆ X : α ∈ A} {Px : x ∈ X‬‬ ‫)ﮐﻪ ‪ A‬ﻣﺠﻤﻮﻋﻪی اﻧﺪﯾﺲﻫﺎﺳﺖ( ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪی ﺷﻤﺎرای ‪ {xn : n ∈ N} ⊆ X‬از ﻋﻨﺎﺻﺮ ‪ X‬وﺟﻮد‬ ‫داﺷﺘﻪ ﺑﺎﺷﺪ‪ ،‬ﺑﻪﻃﻮری ﮐﻪ ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬ﮐﻪ ‪ k ∈ N ،Oα ( Oβ‬ﻣﻮﺟﻮد ﺑﺎﺷﺪ ﮐﻪ ⊆ ‪Oα ⊆ Ox⋆k‬‬ ‫‪ Oβ‬و ‪ Ox⋆k‬ﻣﺠﻤﻮﻋﻪی ﻏﯿﺎب ‪ xk‬ﻧﺴﺒﺖ ﺑﻪ زﯾﺮﺧﺎﻧﻮادهی }‪O = {Oα ⊆ X : α ∈ A‬از ‪ S‬اﺳﺖ‪.‬‬ ‫‪65‬‬


‫ب( ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ) ‪ (X, T‬در وﯾﮋﮔﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﻧﯿﻢﭘﯿﻮﺳﺘﻪ ‪ SRP‬ﺻﺪق ﻣﯽﮐﻨﺪ اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﺑﺮای‬ ‫ﻫﺮ ﺧﺎﻧﻮادهی ﻣﻮﻟﺪ ﭘﯿﺶﺗﺮﺗﯿﺒﯽ ﻧﯿﻢﭘﯿﻮﺳﺘﻪی‬ ‫}‪F = {Oα ⊆ X : α ∈ A‬‬ ‫)ﮐﻪ ‪ A‬ﻣﺠﻤﻮﻋﻪی اﻧﺪﯾﺲﻫﺎﺳﺖ( ﯾﮏ زﯾﺮﻣﺠﻤﻮﻋﻪی ﺷﻤﺎرای ‪ {xn : n ∈ N} ⊆ X‬از ﻋﻨﺎﺻﺮ‬ ‫‪ X‬وﺟﻮد داﺷﺘﻪ ﺑﺎﺷﺪ‪ ،‬ﺑﻪﻃﻮری ﮐﻪ ﺑﺮای ﻫﺮ ‪ α, β ∈ A‬ﮐﻪ ‪ k ∈ N ،Oα ( Oβ‬ﻣﻮﺟﻮد ﺑﺎﺷﺪ ﮐﻪ‬ ‫‪ Oα ⊆ Ox⋆k ⊆ Oβ‬و ‪ Ox⋆k‬ﻣﺠﻤﻮﻋﻪی ﻏﯿﺎب ‪ xk‬ﻧﺴﺒﺖ ﺑﻪ ﺧﺎﻧﻮادهی ‪ F‬اﺳﺖ‪.‬‬ ‫ﻧﺘﯿﺠﻪ ‪ .١٠.٢‬ﻓﺮض ﮐﻨﯿﺪ ‪ X‬ﯾﮏ ﻓﻀﺎی ﺗﻮﭘﻮﻟﻮژﯾﮏ ﺑﺎ ﺗﻮﭘﻮﻟﻮژی ‪ T‬ﺑﺎﺷﺪ‪ .‬اﮔﺮ ) ‪ (X, T‬ﻫﻤﺒﻨﺪ و ﺟﺪاﯾﯽﭘﺬﯾﺮ ﺑﺎﺷﺪ‪،‬‬ ‫آنﮔﺎه ﻓﻀﺎ در وﯾﮋﮔﯽ ﻧﻤﺎﯾﺶﭘﺬﯾﺮی ﭘﯿﻮﺳﺘﻪ ‪ CRP‬ﺻﺪق ﻣﯽﮐﻨﺪ‪[١].‬‬ ‫ﺗﻮﺟﻪ ‪ .١١.٢‬ﻋﮑﺲ ﻗﻀﯿﻪی ﺑﺎﻻ ﺑﺮﻗﺮار ﻧﯿﺴﺖ‪ .‬ﻣﺠﻤﻮﻋﻪی ﻣﺘﻨﺎﻫﯽ ‪ X‬ﺑﺎ ﺑﯿﺶ از ﯾﮏ ﻋﻨﺼﺮ ﺑﺎ داﺷﺘﻦ ﺗﻮﭘﻮﻟﻮژی‬ ‫ﮔﺴﺴﺘﻪ اﯾﻦ ﻣﻮﺿﻮع را ﻧﺸﺎن ﻣﯽدﻫﺪ؛ زﯾﺮا ﻫﺮ ﺗﺎﺑﻊ ﺳﻮدﻣﻨﺪ روی ‪ X‬ﺑﺎ ﺗﻮﭘﻮﻟﻮژی ﮔﺴﺴﺘﻪ‪ ،‬ﭘﯿﻮﺳﺘﻪ اﺳﺖ و ﻟﺬا ﻫﺮ‬ ‫ﭘﯿﺶﺗﺮﺗﯿﺐ ﮐﻠﯽ ﭘﯿﻮﺳﺘﻪی ‪ -‬روی ‪ X‬ﻧﻤﺎﯾﺶﭘﺬﯾﺮ اﺳﺖ وﻟﯽ ﺑﺎ وﺟﻮد ﺗﻮﭘﻮﻟﻮژی ﮔﺴﺴﺘﻪ روی ‪ ،X‬ﻫﺮ ﺗﮏ ﻧﻘﻄﻪای‬ ‫ﺑﺎز اﺳﺖ و ﻟﺬا ‪ X‬ﻫﻤﺒﻨﺪ ﻧﯿﺴﺖ‪.‬‬ ‫ﻣﺮاﺟﻊ‬ ‫‪1. M. J. Campion, J. C. Candeal, E. Indurain , Preorderable topologies and order-representability‬‬ ‫‪of topological spaces, Appl. Gen. Topol, 2009.‬‬ ‫‪2. D.S. Bridges and G.B. Mehta,Representation of Preference Orderings , Springer, Berlin, 1995.‬‬ ‫)‪3. G. Bosi and G. Herden, On the structure of completely useful topologies, Appl. Gen. Topol.3 (2‬‬ ‫‪(2002), 145–167.‬‬

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B. Najafi, F. Malek and N. H. Kashani :: Contact Finsler structure

Paper No. 1.13

Contact Finsler structure B. Najafi1 , Fereshteh Malek2 and N. H. Kashani3 e-mail: [email protected], Shahed University of Tehran, Tehran, Iran. 2 e-mail: [email protected], K. N. Toosi University of Technology, Tehran, Iran. 3 e-mail: [email protected], K. N. Toosi University of Technology, Tehran, Iran. 1

Abstract In this paper normal almost contact Finsler structures on vector bundles are characterized. We define contact Finsler structures and Sasakian Finsler structures on vector bundles. Then, we prove that every locally symmetric or locally φ−recurrent Sasakian Finsler vector bundle is of φ−flag curvature 1. A 6−dimensional K−contact Finsler structure is Sasakian Finsler structure. Keywords: Almost contact Finsler structure, K−contact, locally symmetric, normal, locally φ−recurrent.

1 Preliminaries Let M be an m dimensional smooth manifold and E(M)=(E,π,M) a smooth vector bundle of rank n. We denote by Vu E the fibre of the vertical bundle V E at u ∈ E and by Hu a complementary space of Vu in the tangent space Tu E. Thus, we have the following decomposition Tu E = Hu ⊕Vu E.

(1.1)

A smooth distribution H = ∪Hu on the vector bundle E(M) is called a non-linear connection. We denote by (xi ,ya ), i = 1,··· ,m, a = 1,··· ,n, the canonical coordinates of a point u ∈ E. Then { ∂∂xi , ∂∂ya } is the natural basis of local vector fields and {dxi ,dya } is its dual basis of local 1-forms on E. Here, we use another local frame for local vector fields on E, { δδxi , ∂∂ya }, and its dual {dxi ,δ ya }, which are adapted to the decomposition (1.1). Indeed, we have ∂ δ ∂ = i − Nia (x,y) a , δ ya = dya + N aj dx j , i δx ∂x ∂y

(1.2)

where Nia are the coefficients of the non-linear connection H. Now, we consider the horizontal and the vertical projectors h and v of H, which are determined by the direct decomposition (1.1). These projectors can be expressed with respect to the adapted bases as h = δδxi ⊗ dxi and v = ∂∂ya ⊗ δ ya . Thus, a vector field X on E can be uniquely written in the form X = X H + X V , where X H = hX and X V = vX

67


are called the horizontal and the vertical components of X, respectively. In the adapted basis, we have δ ∂ X = X i (x,y) i + X a (x,y) a . Hence, δx ∂y X H = X i (x,y)

δ ∂ , X V = X a (x,y) a . i δx ∂y

(1.3)

The non-linear connection H is said to be integrable, if for any two vector fields X and Y on E, we have [X H ,Y H ]v = 0. From now, we suppose that H is integrable. Now, let ω be a 1-form on E. Then it can be uniquely written as ω = ω H + ω V . In the adapted basis, we have ω = ωi (x,y)dxi + ωa (x,y)δ ya . Hence, ω H = ωi (x,y)dxi , ω V = ωa (x,y)δ ya .

(1.4)

A (p + r,q + s)−tensor field T on E is called a distinguished tensor field (briefly, d-tensor) of type p q if it has the following property r s T (ωi1 ,··· ,ωi p ,ωa1 ,··· ,ωar ,X j1 ,··· ,X jq ,Xb1 ,··· ,Xbs ) = T (ωiH1 ,··· ,ωiHp ,ωaV1 ,··· ,ωaVr ,X jH1 ,··· ,X jHq ,XbV1 ,··· ,XbVs ), where ωik , ωal , (k = 1,··· , p, l = 1,··· ,r) are 1-forms on E and X jv ,Xbw , (v = 1,··· ,q, w = 1,··· ,s) are vector fields on E. Thus, T is expressed by ∂ δ ∂ δ ⊗ ··· ⊗ i p ⊗ a ⊗ ··· ⊗ a i 1 1 δx ∂y ∂y r δx ⊗dx j1 ⊗ ··· ⊗ dx jq ⊗ δ yb1 ⊗ ··· ⊗ δ ybs . i ,···,i ,a ,···,a

T = T j11,···, jqp,b11,···,bsr

(1.5)

A linear connection D on the manifold E is called a distinguished connection (briefly, d−connection), if it preserves the horizontal distribution, (i.e., Dh = 0), and consequently Dv = 0 due to Id = h + v. Therefore, we can write DX Y = (DX Y H )H + (DX Y V )V , DX ω = (DX ω H )H + (DX ω V )V , where X,Y are vector fields and ω is a 1−form on E. Define DHX Y = DX H Y and DVX Y = DX V Y . Then, a d−connection D, with respect to an adapted basis, locally is given by the following δ δ ∂ ∂ a = Fjki (x,y) i , DHδ = F (x,y) , bk j δx δx ∂ ya ∂ yb δ xk δ xk

DHδ

δ ∂ δ ∂ = V jai (x,y) i , DV = Vbca (x,y) a . b ∂ δxj ∂ δx ∂y ∂y a c ∂y ∂y

DV

68

(1.6)


It is well known that DH and DV act as covariant derivative in the algebra of d−tensor fields on E. We call DH (resp. DV ) the operator of h−covariant (resp. v−covariant) derivation. If ω is a 1−form on E, we define for any vector fields X,Y on E (DHX ω)(Y ) = X H (ω(Y )) − ω(DHX Y ), Let T stands for the torsion of a d−connection D on E. Then T is completely determined by the following five tensor fields: T H (X H ,Y H ) = DHX Y H − DYH X H − [X H ,Y H ]H , T V (X H ,Y H ) = −[X H ,Y H ]V , T H (X H ,Y V ) = −DVY X H − [X H ,Y V ]H , T V (X H ,Y V ) = DHX Y V − [X H ,Y V ]V , T V (X V ,Y V ) = DVX Y Y − DVY X V − [X V ,Y V ]V , which are called (h)h− torsion, (v)h− torsion, (h)hv− torsion, (v)hv−torsion and (v)v−torsion, respectively. Due to integrability of the non-linear connection H, (v)h− torsion vanishes. A d−connection D is said to be symmetric if the (h)h−torsion and (v)v−torsion vanish. It is known that the exterior differential of a q−form on E, say ω, can be given in terms of any linear connection D on E and its torsion tensor T as follows q+1

dω(X1 ,...,Xq+1 ) =

∑ (−1)i+1(DXi ω)(X1,..., Xei,...,Xq+1)

i=1

−

∑

(−1)i+ j ω(T (Xi ,X j ),X1 ,..., Xei ,..., Xej ,...,Xq+1 ),

1≤i< j≤q+1

where the tilde sign above a term means omitting that term. Proposition 1.1. ([4]) If ω is a 1−form and D is a d-connection on E, then we have the following dω(X H ,Y H ) = (DHX ω)Y H − (DYH ω)X H + ω(T (X H ,Y H )).

(1.7)

dω(X V ,Y H ) = (DVX ω)Y H − (DYH ω)X V + ω(T (X V ,Y H )).

(1.8)

dω(X V ,Y V ) = (DVX ω)Y V − (DVY ω)X V + ω(T (X V ,Y V )).

(1.9)

We consider a pseudo-metric G on the manifold Ebeing symmetric and non-degenerate, as G = GH +GV , 0 2 where GH (X,Y ) = G(X H ,Y H ) is of type , symmetric and non-degenerate on H and GV (X,Y ) = 0 0 0 0 V V G(X ,Y ) is of type , symmetric and non-degenerate on V E. In the adapted basis, we can write 0 2 G = gi j (x,y)dxi ⊗ dx j + gab (x,y)δ ya ⊗ δ yb .

69


A d-connection D on E is called a metrical d-connection with respect to G if DX G = 0 holds for every vector field X on E. In the sequel, we consider symmetric metrical d−connections and call them Finsler connections. Finally, we consider the curvature of a Finsler connection D defined by R(X,Y )Z = DX DY Z − DY DX Z − D[X,Y ] Z, ∀X,Y,Z ∈ χ(E), where χ(E) is the set of all vector fields on E. As D preserves the horizontal and the vertical distributions, so dose R(X,Y ). Consequently, we have the following R(X,Y )Z = (R(X,Y )Z H )H + (R(X,Y )ZV )V , ∀X,Y,Z ∈ χ(E), from which we conclude that R(X,Y ) is completely determined by the following six tensor fields R(X H ,Y H )Z H = DHX DYH Z H − DYH DHX Z H − D[X H ,Y H ] Z H , R(X H ,Y H )ZV = DHX DYH ZV − DYH DHX ZV − D[X H ,Y H ] ZV , R(X V ,Y H )Z H = DVX DYH Z H − DYH DVX Z H − D[X V ,Y H ] Z H , R(X V ,Y H )ZV = DVX DYH ZV − DYH DVX ZV − D[X V ,Y H ] ZV , R(X V ,Y V )Z H = DVX DVY Z H − DVY DVX Z H − D[X V ,Y V ] Z H , R(X V ,Y V )ZV = DVX DVY ZV − DVY DVX ZV − D[X V ,Y V ] ZV . We call the first and the sixth equations as horizontal curvature and vertical curvature of D, respectively.

2

Contact Finsler Vector Bundle

Let E(M)=(E,π,M) be a smooth vector bundle with an integrable non-linear connection H on E(M). A triplet (φ,η,ξ ) is named an almost contact Finsler structure on E(M), if the following conditions hold φ 2 = −I + η H ⊗ ξ H + η V ⊗ ξ V ,

(2.1)

η H (ξ H ) = η V (ξ V ) = 1,

(2.2)

where φ is a (1,1)−tensor field, η is a 1-form and ξ is a vector field on E. Moreover, φ H and φ V are defined by φ H (X) = φ(X H ) and φ V (X) = φ(X V ). In this case E(M) is said to be an almost contact Finsler vector bundle ([5]). A pseudo-metric structure G on E satisfying the conditions GH (φX,φY ) = GH (X,Y ) − η H (X H )η H (Y H ),

(2.3)

GV (φX,φY ) = GV (X,Y ) − η V (X V )η V (Y V ),

(2.4)

70


is said to be compatible with the structure (φ,ξ ,η). In this case, the tetrad (φ,ξ ,η,G) is called an almost contact metric Finsler structure and E(M) is called an almost contact metric Finsler vector bundle. From (2.1), (2.2), (2.3) and (2.4) we deduce G(φX,φY ) = G(X,Y ) − η H (X H )η H (Y H ) − η V (X V )η V (Y V ),

(2.5)

GH (X,ξ ) = η H (X), GV (X,ξ ) = η V (X).

(2.6)

Using (2.1), one can also obtain GH (φX,φY ) = −GH (φ 2 X,Y ), GV (φX,φY ) = −GV (φ 2 X,Y ).

(2.7)

The fundamental 2−form Φ is defined by ΦH (X,Y ) = GH (X H ,φ(Y H )), ΦV (X,Y ) = GV (X V ,φ(Y V )).

(2.8)

Proposition 2.1. Let E(M) be an almost contact Finsler vector bundle. Then E is of even dimension. Let (E(M),φ,ξ ,η) be an almost contact Finsler vector bundle. Define J1 and J2 on E as follows J1 = φ − η H ⊗ ξ V + η V ⊗ ξ H , J2 = φ + η H ⊗ ξ V − η V ⊗ ξ H .

(2.9) (2.10)

It is easy to see that J1 and J2 are almost complex Finsler structures on E. If the Nijenhuis torsion of J1 and J2 vanish, we say that (E(M),φ,ξ ,η) is normal. It is natural to give a characterization of normality of E(M) in terms of (φ,ξ ,η). For this, define the tensor N (1) as follows N (1) (X H ,Y H ) = Nφ (X H ,Y H ) + dη H (X H ,Y H )ξ H ,

(2.11)

N (1) (X V ,Y V ) = Nφ (X V ,Y V ) + dη V (X V ,Y V )ξ V ,

(2.12)

N (1) (X V ,Y H ) = Nφ (X V ,Y H ) + dη V (X V ,Y H )ξ V + dη H (X V ,Y H )ξ H .

(2.13)

Theorem 2.2. An almost contact Finsler vector bundle (E(M),φ,ξ ,η) is normal if and only if N (1) = 0. Definition 2.3. By a contact Finsler vector bundle we mean a vector bundle together with a 1−form η such that η H ∧ η V ∧ (dη)n 6= 0

(2.14)

Definition 2.4. Let (φ,η,ξ ,G) be a contact metrical Finsler structure on E . If the metric structure of E is normal, then E is mentioned to have a Sasakian Finsler structure. It is known that an almost contact metrical Finsler vector bundle (E(M),φ,ξ ,η) is a Sasakian Finsler vector bundle if and only if ([5]) (∇HX φ)Y = GH (X,Y )ξ H − η H (X)Y H

(2.15)

(∇VX φ)Y

(2.16)

V

V

V

= G (X,Y )ξ − η (X)Y

71

V


Proposition 2.5. Let (φ,ξ ,η,G) be a Sasakian Finsler structure on a vector bundle E(M). Then we have ∇HX ξ H = −φX H , ∇VX ξ V = −φX V ,

(2.17)

∇HX ξ V = 0, ∇VX ξ H = 0. (∇HX η H )Y H = GH (X H ,φY H ), (∇VX η V )Y V (∇HX η V )Y V = 0, (∇VX η H )Y H = 0. H H H H V

(2.18) = GV (X V ,φY V ),

(2.20) V

V

R(X,Y )ξ = η (Y )X − η (X)Y + η (Y )X − η (X)Y H

H

H

H

(2.19)

V

V

V

V

(2.21)

V

η(R(X,Y )Z) = G(Y ,Z )η (X ) + G(Y ,Z )η (X ) − G(X H ,Z H )η H (Y H ) − G(X V ,ZV )η V (Y V )

(2.22)

A Sasakian Finsler vector bundle is said to be a locally φ− symmetric, if for all vector fields X,Y,Z,W orthogonal to ξ H and ξ V the following holds φ 2 ((∇W R)(X,Y )Z) = 0,

(2.23)

and it is said to be a locally φ−recurrent, if there exists a non-zero 1−form B such that φ 2 ((∇W R)(X,Y )Z) = B(W )R(X,Y )Z,

(2.24)

for all X,Y,Z,W orthogonal to ξ H and ξ V . Let (E(M),φ,ξ ,η) be a Sasakian Finsler vector bundle, it is proved that the following relations hold (∇Z R)(X,Y )ξ =φR(X,Y )Z + GH (X,Z)φY H + GV (X,Z)φX V H

H

V

− G (Y,Z)φX − G (Y,Z)φX

V

(2.25) (2.26)

Theorem 2.6. Let (φ,η,ξ ,G) be a Sasakian Finsler structure on E. Suppose that E is locally symmetric or locally φ−recurrent. Then it has φ−flag curvature 1. Theorem 2.7. A 6−dimensional K−contact Finsler vector bundle is Sasakian Finsler vector bundle.

References [1] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhuser, Basel, 2002. [2] K. Kenmotsu, A class of contact Riemannian manifold, Tohoku Math. Journal 24 (1972), 93103. [3] R. Miron, Vector bundle Finsler geometry, Proc. Nat Sem. on Fisnler spaces-2, Brasov (1982), 147-186. [4] B. B. Sinha and R. K. Yadav, An almost contact Finsler structures on vector bundle, Indian J. Pure. Appl. Math, 19(1) (1988), 27-35. [5] A. F. Yalinz and N. Caliskan, Sasakian Finsler manifolds, Turk. J. Math., (2010), 1-22 .

72

M. Nadjafikhah and K. Goodarzi :: Integrating factor and λ-symmetry for third-order differential equations

Paper No. 1.14

INTEGRATING FACTOR AND λ -SYMMETRY FOR THIRD-ORDER DIFFERENTIAL EQUATIONS Mehdi Nadjafikhah1 and KHodayar Goodarzi2 1 e-mail: m [email protected], Iran University of Science and Technology, Tehran, Iran. 2 e-mail: [email protected], Islamic Azad University, Karaj, Tehran, Iran. Abstract In this paper, we will obtain first integral, integrating factor, λ -symmetry of third-order ODEs ... u = F(x,u, u, ˙ u) ¨ and the relationship between them. Keywords: Symmetry, λ -Symmetry, Integrating factor, First integral, Order reduction.

1 Introduction There are many examples of ODEs that have trivial Lie symmetries. In 2001, Muriel and Romero introducted λ -symmetry to find general solutions for such examples. Recently, they [7] presented techniques to obtain first integral, integrating factor, λ -symmetry of second-order ODEs u¨ = F(x,u, u) ˙ and the relationship between them. In this paper, we will obtain first integral, integrating factor, λ -symmetry of ... third-order ODEs u = F(x,u, u, ˙ u) ¨ and the relationship between them.

2 λ -symmetries on ODEs In this section we recall some of the foundational results about λ -symmetry rather briefly. An integrating factor of an nth-order ordinary differential equation (ODE) ∆(x,u(n) ) = 0

(2.1)

is a function µ (x,u(n−1) ) such that the equation µ ∆ = 0 is an exact equation:

µ (x,u(n−1) )∆(x,u(n) ) = Dx (G(x,u(n−1) ))

(2.2)

function G(x,u(n−1) ) in (2.2) will be called a first integral of the Eq. (2.1) and Dx (G(x,u(n−1) )) = 0 is a conserved form of the Eq. (2.1). Let u(n) = F(x,u(n−1) ) (2.3)

73


be a nth-order ordinary differential equation, where F is an analytic function of its arguments. If λ (x,u(k) ), for some k < n, is any particular solution (Dx + λ )n (1) =

n−1

∂F

∑ (Dx + λ )i(1) ∂ ui

(2.4)

i=0

then the vector field v = ∂u is a λ -symmetry of Eq. (2.3). For n = 3, the corresponding third-order ODEs can be written in explicit form as particular ... u = F(x,u, u, ˙ u). ¨

(2.5)

˙ u) ¨ ∂u¨ the vector field associated with (2.5). Function We denote by A = ∂x + u˙∂u + u¨∂u˙ + F(x,u, u, I(x,u, u, ˙ u) ¨ is a first integral such that A(I) = 0 and an integrating factor is any function µ (x,u, u, ˙ u) ¨ ... such that µ ( u − F(x,u, u, ˙ u)) ¨ = Dx (I) of the (2.5) and also Dx is the total derivative vector field: Dx = ∂x + u˙∂u + u¨∂u˙ + ... . If λ (x,u, u, ˙ u) ¨ is any particular solution of D2x λ + Dx λ 2 + λ Dx λ + λ 3 =

∂F ∂F ∂F + λ + (Dx λ + λ 2 ) ∂u ∂ u˙ ∂ u¨

then the vector field v = ∂u is a λ -symmetry of Eq. (2.5). ˙ u) ¨ = Iu¨ (x,u, u, ˙ u) ¨ is a integrating Theorem 2.1. If I(x,u, u, ˙ u) ¨ is a first integral of Eq. (2.5), then µ (x,u, u, factor of (2.5). Proof. If I(x,u, u, ˙ u) ¨ be a first integral of Eq. (2.5), then 0 = A(I) = Ix + uI ˙ u + uI ¨ u˙ + F(x,u, u, ˙ u)I ¨ u¨ , therefore Ix + uI ˙ u + uI ¨ u˙ = −F(x,u, u, ˙ u)I ¨ u¨ and ... ... ... ˙ u)). ¨ Dx I = Ix + uI ˙ u + uI ¨ u˙ + u Iu¨ = −F(x,u, u, ˙ u)I ¨ u¨ + u Iu¨ = Iu¨ ( u − F(x,u, u, Hence µ = Iu¨ . Theorem 2.2. If µ (x,u, u, ˙ u) ¨ is an integrating factor of Eq. (2.5), then there is a first integral I(x,u, u, ˙ u) ¨ ˙ u) ¨ = Iu¨ (x,u, u, ˙ u). ¨ of Eq. (2.5), such that µ (x,u, u, ... ˙ u) ¨ is an integrating factor of Eq. (2.5), then µ (x,u, u, ˙ u)( ¨ u − F(x,u, u, ˙ u)) ¨ = Dx (I) = Proof. If µ (x,u, u, ... Ix + uI ˙ u + uI ¨ u˙ + u Iu¨ , for some function I(x,u, u, ˙ u) ¨ then µ (x,u, u, ˙ u) ¨ = Iu¨ (x,u, u, ˙ u) ¨ also, we have −µ (x,u, u, ˙ u)F(x,u, ¨ u, ˙ u) ¨ = −Iu¨ F(x,u, u, ˙ u) ¨ = Ix + uI ˙ u + uI ¨ u˙ therefore Ix + uI ˙ u + uI ¨ u˙ + F(x,u, u, ˙ u)I ¨ u¨ = 0, i.e. A(I) = 0.

74


If v[λ ,(k)] (α (x,u(k) )) = v[λ ,(k)] (β (x,u(k) )) = 0 where α (x,u(k) ), β (x,u(k) ) ∈ C∞ (M (k) ) then v[λ ,(k+1)]

( D α (x,u(k) ) ) x = 0. Dx β (x,u(k) )

Theorem 2.3. If I(x,u, u, ˙ u) ¨ is a first integral of Eq. (2.5), then the vector field v = ∂u is a λ -symmetry of (2.5) such that λ is solution Iu + λ Iu˙ + (Dx λ + λ 2 )Iu¨ = 0 and v[λ ,(2)] (I) = 0. Proof. Since for any function λ (x,u, u, ˙ u), ¨ we have v[λ ,(2)] = ∂ u + λ ∂ u˙ + (Dx λ + λ 2 )∂ u,¨ therefore, v[λ ,(2)] (I) = Iu + λ Iu˙ + (Dx λ + λ 2 )Iu¨ = 0. Dx I Since functions g(x,u, u, ˙ u) ¨ = x and I(x,u, u, ˙ u) ¨ are first integral of v[λ ,(2)] then v[λ ,(3)] ( ) = v[λ ,(3)] (Dx I) = Dx x ... [ λ ,(3)] [ λ ,(3)] u 0, i.e., Dx I is an invariant of v . By applying v to identity µ ( −F(x,u, u, ˙ u)) ¨ = Dx (I), we obtain ) ( ... ˙ u)) ¨ = v[λ ,(3)] (Dx (I)) v[λ ,(3)] µ ( u − F(x,u, u, ( ... ) ˙ u)) ¨ = 0 v[λ ,(3)] Iu¨ ( u − F(x,u, u, (... ) (... ) v[λ ,(3)] (Iu¨ ) u − F(x,u, u, ˙ u) ¨ + Iu¨ v[λ ,(3)] u − F(x,u, u, ˙ u) ¨ = 0 (... ) ... Iu¨ v[λ ,(3)] u − F(x,u, u, ˙ u) ¨ = 0 when u = F(x,u, u, ˙ u) ¨ since Iu¨ ̸= 0, hence the vector field v = ∂u is a λ -symmetry of (2.5). The vector field v = ξ (x,u)∂x + η (x,u)∂u is a λ -symmetry of equation (2.5) if and only if [v[λ ,(2)] ,A] = λ v[λ ,(2)] + τ A where τ = −(A + λ )(ξ (x,u)). When v = ∂u is a λ -symmetry of equation (2.5) if and only if [v[λ ,(2)] ,A] = λ v[λ ,(2)] . ˙ u), ¨ then there is a first Theorem 2.4. If v = ∂u is a λ -symmetry of (2.5) for some function λ (x,u, u, [ λ ,(2)] (I) = 0. integral I(x,u, u, ˙ u) ¨ of (2.5) such that v Proof. If v = ∂u is a λ -symmetry of (2.5) for some function λ (x,u, u, ˙ u), ¨ then [v[λ ,(2)] ,A] = λ v[λ ,(2)] . Therefore {v[λ ,(2)] ,A} is an involutive set of vector fields in M (2) and there is function I(x,u, u, ˙ u) ¨ such [ λ ,(2)] that v (I) = 0 and A(I) = 0. Suppose ω (x,u, u, ˙ u) ¨ be a first integral of v[λ ,(2)] = ∂ u + λ ∂ u˙ + (Dx λ + λ 2 )∂ u,¨ i.e., ω (x,u, u, ˙ u) ¨ is a solution of the second-order PDE

ωu + λ ωu˙ + (Dx λ + λ 2 )ωu¨ = 0.

75


If I(x,u, u, ˙ u) ¨ = G(x, ω (x,u, u, ˙ u)) ¨ be a first integral (2.5), then 0 = = = =

A(I) ... Ix + uI ˙ u + uI ¨ u˙ + u Iu¨ Ix + uI ˙ u + uI ¨ u˙ + F(x,u, u, ˙ u)I ¨ u¨ (Gx + Gω ωx ) + u(G ˙ ω ωu ) + u(G ¨ ω ωu˙ ) + F(x,u, u, ˙ u)(G ¨ ω ωu¨ ) ( ) ˙ u) ¨ ωu¨ Gω = Gx + ωx + u˙ωu + u¨ωu˙ + F(x,u, u, = Gx + A(ω )Gω = Gx + H(x, ω )Gω

where A(ω ) in terms of (x, ω ) as A(ω ) = H(x, ω ). Hence, if G(x, ω ) be a solution of Gx +H(x, ω )Gω = 0, ˙ u)) ¨ is a first integral of (2.5). then I(x,u, u, ˙ u) ¨ = G(x, ω (x,u, u, In summary, a procedure to find a first integral I(x,u, u, ˙ u) ¨ and consequently an integrating factor of (2.5) is as follows: • The vector field v = ∂u is a λ -symmetry of (2.5), if function λ (x,u, u, ˙ u) ¨ be any particular solution of the equation D2x λ + Dx λ 2 + λ Dx λ + λ 3 =

∂F ∂F ∂F + λ + (Dx λ + λ 2 ) ∂u ∂ u˙ ∂ u¨

(2.6)

• Find a first integral ω (x,u, u, ˙ u) ¨ of v[λ ,(2)] ,i.e., a particular solution of the equation

ωu + λ ωu˙ + (Dx λ + λ 2 )ωu¨ = 0.

(2.7)

• Evaluate A(ω ) = H(x, ω ). • Find a first integral G(x, ω ) from solve of the equation Gx + H(x, ω )Gω = 0.

(2.8)

• The function I(x,u, u, ˙ u) ¨ = G(x, ω (x,u, u, ˙ u)) ¨ is a first integral of (2.5). • The function µ (x,u, u, ˙ u) ¨ = Iu¨ (x,u, u, ˙ u) ¨ is an integrating factor of (2.5).

3 Example Consider the third-order differential equation ) ... ( u − f1 (x)u¨ + f2 (x)u˙ + f3 (x)u + f4 (x) = 0

76

(3.1)


where fi (x), i = 1,2,3,4 are arbitrary functions and F(x,u, u, ˙ u) ¨ = f1 (x)u¨ + f2 (x)u˙ + f3 (x)u + f4 (x) is an 1 analytic function of its arguments. It can be checked that λ = is a particular solution of (2.6), where x f2 (x) f1 (x), f2 (x), f4 (x) are arbitrary functions and f3 (x) = − . To find an integrating factor associated x to λ , first, we find a first integral invariant ω (x,u, u, ˙ u) ¨ of v[λ ,2] by the equation that corresponds to (2.7), which means, 1 1 1 1 ωu + ωu˙ + (Dx ( ) + ( )2 )ωu¨ = ωu + ωu˙ = 0. (3.2) x x x x u The solution of this equation is ω = u¨ + u˙ − . The vector field associated A = ∂x + u˙∂u + u¨∂u˙ + x F(x,u, u, ˙ u) ¨ ∂u¨ acts on ω , then, we have u˙ u ... A(ω ) = u + u¨ − + 2 x x ( ) u˙ u = f1 (x)u¨ + f2 (x)u˙ + f3 (x)u + f4 (x) + u¨ − + 2 x x ( ) u˙ u f2 (x) u + f4 (x) + u¨ − + 2 = f1 (x)u¨ + f2 (x)u˙ − x x x 1 u = ( f1 (x) + 1)u¨ + ( f2 (x) − )(u˙ − ) + f4 (x) x x u = ( f1 (x) + 1)(u¨ + u˙ − ) = ( f1 (x) + 1)ω = H(x, ω ). x 1 where f2 (x) = f1 (x) + + 1 and f4 (x) = 0. Therefore, A(ω ) = ( f1 (x) + 1)ω = H(x, ω ). The function x ( ∫ ) G(x, ω ) = ω exp − ( f1 (x) + 1)dx is a particular solution for the equation that corresponds to (2.8), i.e., Gx + (( f1 (x) + 1)ω )Gω = 0. Therefore, ( ∫ ) u ˙ u)) ¨ = (u¨ + u˙ − )exp − ( f1 (x) + 1)dx I(x,u, u, ˙ u) ¨ = G(x, ω (x,u, u, x is a first integral of (2.5). According to Theorem 2.1, the function ( ∫ ) µ (x,u, u, ˙ u) ¨ = Ix¨(x,u, u, ˙ u) ¨ = exp − ( f1 (x) + 1)dx is an integrating factor of (2.5). Finally, we have ) ( (... ( ∫ ) f2 (x) ) u u = Dx (G(x, ω )) = Dx (u¨ + u˙ − )exp − ( f1 (x) + 1)dx = 0. µ . u − f1 (x)u¨ − f2 (x)u˙ + x x f2 (x) ... Therefore, we reduce the order of equation u − f1 (x)u¨ − f2 (x)u˙ + u = 0 to the equation x u u¨ + u˙ − = 0. x

77


4 First integral, Integrating factor and λ -symmetry Theorem 4.1. A system of the form  ( ) 2 )u˙ − H u¨  I = µ − F + ( λ H + D λ + λ  x x   ( )  2 Iu = −µ λ H + Dx λ + λ   Iu˙ = µ H    Iu¨ = µ

(4.1)

is compatibly for some function λ (x,u, u, ˙ u) ¨ , µ (x,u, u, ˙ u) ¨ and H(x,u, u, ˙ u), ¨ if and only if µ is an integrating factor of (2.5) and v = ∂u is a λ -symmetry of (2.5). In this case I is a first integral of (2.5). Proof. If I be a first integral of (2.5) then µ = Iu¨ is an integrating factor of (2.5) and if v = ∂u be a λ -symmetry of (2.5) then A(I) = 0 and v[λ ,(2)] (I) = 0, i.e. Ix = −uI ˙ u − uI ¨ u˙ − FIu¨ = −uI ˙ u − uI ¨ u˙ − F µ 2 Iu = −λ Iu˙ − (Dx λ + λ )Iu¨ = −λ Iu˙ − (Dx λ + λ 2 )µ . If Iu˙ = µ H, where H(x,u, u, ˙ u) ¨ is arbitrary function, then system (4.1) is compatible. We are going to prove that, when (4.1) is compatible necessarily v = ∂u is a λ -symmetry. Suppose (4.1) is compatible,i.e., Ixu = Iux ,Ixu˙ = Iux˙ ,Ixu¨ = Iux¨ ,Iuu˙ = Iuu ˙ ,Iuu¨ = Iuu ¨ ,Iu˙u¨ = Iuü˙ . Obviously that Ixu¨ = Iux ¨ ,Iuu¨ = Iuu ¨ ,Iu˙u¨ = Iuü˙ , implies that ( ) ( 2 µx = (Iu¨ )x = (Ix )u¨ = µu¨ − F + (λ H + Dx λ + λ )u˙ − H u¨ + µ − F + (λ H + Dx λ + λ 2 )u˙ ) − H u¨ u¨ ( ) ( ) 2 2 µu = (Iu¨ )u = (Iu )u¨ = −µu¨ λ H + Dx λ + λ − µ λ H + Dx λ + λ (4.2) u¨

µu˙ = (Iu¨ )u˙ = (Iu˙ )u¨ = µu¨ H + µ Hu¨ . The compatibility of system (4.1) and by using of (4.2) implies that

A(λ H + Dx λ + λ 2 ) − Fu − (Fu¨ + H)(λ H + Dx λ + λ 2 ) = 0 −A(H) − Fu˙ + HFu¨ + H 2 + λ H + Dx λ + λ 2 = 0 A(µ ) + µ Fu¨ + µ H = 0.

78

(4.3) (4.4) (4.5)


By using (4.3) we have [ ] 0 = µ A(λ H + Dx λ + λ 2 ) − Fu − (Fu¨ + H)(λ H + Dx λ + λ 2 ) [ ] 2 2 2 2 = µ HDx λ + λ Dx H + Dx λ + Dx λ − Fu − λ HFu¨ − (Dx λ + λ )Fu¨ − H(λ H + Dx λ + λ ) ) [ ( = µ HDx λ + λ − Fu˙ + HFu¨ + H 2 + λ H + Dx λ + λ 2 + D2x λ + Dx λ 2 − Fu − λ HFu¨ ] − (Dx λ + λ 2 )Fu¨ − H(λ H + Dx λ + λ 2 ) ] [ = µ D2x λ + Dx λ 2 + λ Dx λ + λ 3 − Fu − λ Fu˙ − (Dx λ + λ 2 )Fu¨ . Hence,

[ ] µ D2x λ + Dx λ 2 + λ Dx λ + λ 3 − Fu − λ Fu˙ − (Dx λ + λ 2 )Fu¨ = 0

(4.6)

when µ ̸= 0, (4.6) implies that v = ∂u is a λ -symmetry. In summary, a procedure to find an integrating factor µ (x,u, u, ˙ u) ¨ and consequently a first integral I(x,u, u, ˙ u) ¨ of (2.5) is as follows: ˙ u) ¨ be any particular solution • The vector field v = ∂u is a λ -symmetry of (2.5), if function λ (x,u, u, of the equation D2x λ + Dx λ 2 + λ Dx λ + λ 3 =

∂F ∂F ∂F + λ + (Dx λ + λ 2 ) ∂u ∂ u˙ ∂ u¨

(4.7)

• By using of (4.3)-(4.5), we find an integrating factor µ (x,u, u, ˙ u) ¨ and function H(x,u, u, ˙ u) ¨ of (2.5), i.e., a particular solution of the system   A(λ H + Dx λ + λ 2 ) − Fu − (Fu¨ + H)(λ H + Dx λ + λ 2 ) = 0 (4.8) −A(H) − Fu˙ + HFu¨ + H 2 + λ H + Dx λ + λ 2 = 0  A(µ ) + µ Fu¨ + µ H = 0. • Find a first integral I(x,u, u, ˙ u) ¨ from solve of the system  ) ( 2  Ix = µ (λ H + Dx λ + λ )u˙ − H u¨ − F    ) (  Iu = −µ λ H + Dx λ + λ 2   Iu˙ = µ H    Iu¨ = µ • We have

(... ) ( ) µ (x,u, u, ˙ u) ¨ u − F(x,u, u, ˙ u) ¨ = Dx I(x,u, u, ˙ u) ¨ .

79

(4.9)


5 Example Consider the third-order differential equation 2x 3x2 ... u − 2 u¨ − 2 =0 x +1 x +1

(5.1)

2x 3x2 u ¨ + is an analytic function of its arguments. It can be checked that x2 + 1 x2 + 1 1 1 2x 3x2 λ = is a particular solution of (4.7). Substituting F(x,u, u, ˙ u) ¨ = 2 u¨ + 2 and λ = into (4.8) x x +1 x +1 x and solving them, we obtain H = 0 and µ = x2 + 1. Therefore, by using of system (4.9), we have  Ix = −µ F = 2xu¨ + 3x2    Iu = 0 (5.2) Iu˙ = 0    Iu¨ = µ = x2 + 1. (... A solution of this system is I(x,u, u, ˙ u) ¨ = (x2 + 1)u¨ + x3 . Therefore, by using of (2.2), i.e. (x2 + 1) u − ( ) 2x 2x 3x2 ) ... 2 3 = Dx (x + 1)u¨ + x , implies that, we reduce the order of equation u − 2 u¨ − u¨ − 2 2 x +1 x +1 x +1 3x2 = 0 to the equation (x2 + 1)u¨ + x3 = 0. x2 + 1 where F(x,u, u, ˙ u) ¨ =

References [1] P.J. O LVER, Applications of Lie Groups to Differential Equations, (New York, 1986). [2] B. A BRAHAM -S HRDAUNER, Hidden symmetries and non-local group generators for ordinary differential equation, IMA J. Appl. Math. 56 (1996)235–252. [3] C.M URIEL and J. L. ROMERO, New methods of reduction for ordinary differential equation, IMA J. Appl. Math. 66 (2001) 111–125. [4] C. M URIEL and J. L. ROMERO, C∞ -symmetries and reduction of equation without Lie point symmetries, J. Lie Theory, 13 (2003) 167–188. [5] G. G AETA and P. M ORANDO, On the geometry of lambda-symmetries and PDEs reduction, J. Phys. A,37 (2004) 6955–6975. [6] C. M URIEL and J. L. ROMERO, λ -symmetries and integrating factors, J. Non-linear Math. Phys. 15 (2008) 290–299.

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[7] C. M URIEL and J. L. ROMERO, First integrals, integrating factors and λ -symmetries of secondorder differential equations, J. Phys. A:Math. Theor.43 (2009)365207. [8] C. M URIEL and J. L. ROMERO, Second-order ordinary differential equations and first integral of the form A(t,x)x˙ + B(t,x) , J. Nonlinear Math. Phys.16 (2009) 209–222. [9] C. M URIEL and J. L. ROMERO, A λ -symmetry-based method for the linearization and determination of first-integrals of a family of second-order differential equations, J. Phys. A:Math. Theor.44 (2011)245201. [10] E. YASAR, Integrating factors and first integral for Lienard type and frequency-damped oscillators, Mathematical Problems in Engineering Volume 2011, Article ID 916437, 10 pages doi:10.1155/2011/916437. (2011).

81

R. Mirzaie and H. Soruosh :: A remark on topology of some UND-manifolds

Paper No. 1.15

A remark on topology of some UND-manifolds 1

R. Mirzaie and H. Soroush Department of Mathematics, Faculty of Sciences I. Kh. International University, Qazvin, Iran. e-mail: [email protected], 2 Department of Mathematics, Payame Noor Universtiy, PO BOX 19395-3697 Tehran,IRAN e-mail: [email protected], Abstract We classify cohomogeneity two UND-Riemannian G-manifolds under the condition M G ̸= 0. /

1. Introduction A classic theorem about Riemannian manifolds of non-positive curvature ([16]) states that a homogeneous Riemannian manifold M of non positive curvature is simply connected or it is diffeomorphic to a cylinder over a torus (i.e, it is diffeomorphic to Rk × T s , k + s = dimM). It is interesting to reduce the homogeneity condition to weaker conditions and see what happens to the topology of M. When M is homogeneous then there is a connected and closed subgroup G of the isometries of M such that M the orbit space of the action of G on M, M G is a one point set. A weaker condition is that dim G = 1 or 2 (i.e, M be a cohomogeneity one or cohomogeneity two G-manifold). There are some interesting theorems about topological properties of cohomogeneity one G-manifolds of non-positive curvature under conditions on G and M (see [1],[12], [13], [15]). There is a topological characterization of cohomogeneity one UND-Riemannian manifolds ( Riemannian manifolds with the property that the universal covering manifold decomposes as a direct product of negatively curved manifolds) in [13]. Following the papers [9-11], where the first author proved various results about topological properties of cohomogeneity two negatively curved G-manifold M under some special conditions on M or G, we are going to consider some cohomogeneity two UND- manifolds in the present paper. We topologically characterize a UND-manifold M which is acted on isometrically by a connected and closed subgroup G of isometries, under the condition that the fixed point set of the action is not empty. Our main result is Theorem 3.5.

∗ MSC(2000): 53C30, 57S25. Keywords: Lie group; Isometry; Manifold.

82


2. Preliminaries We will use the following definitions, facts and symbols in the proof of the main theorem. Definition 2.1. A differentiable function f : M → R on a complete Riemannian manifold M is called convex ( strictly convex) if for each geodesic γ : R → M the composed function f ◦ γ : R → R is convex ( strictly convex), that is ( f oγ )′′ ≥ 0 ( ( f ◦ γ )′′ > 0). Fact 2.2 (see [2], [5]). (1) Let M be a simply connected Riemannian manifold of nonpositive curvature and δ ∈ Iso(M). The squared displacement function dδ2 : M → M defined by dδ2 (x) = (d(x, δ x))2 is a convex function. (2) If in (1) M has negative curvature then dδ2 is strictly convex except at the minimum point set. (3) Let M be a simply connected Riemannian manifold of negative curvature and δ be an isometry on M without fixed point. If there is a geodesic γ such that δ (γ ) = γ then the image of γ is the minimum point set of the function dδ2 : M → M. (4) If f : M → R is a convex function defined on a complete Riemannian manifold M then the minimum point set C of f is a totally convex subset of M (i.e, it contains every geodesic segment with endpoints inside C). (5) Let M be a complete Riemannian manifold of nonpositive curvature. A submanifold S of M is closed and totally convex if and only if S is totally geodesic and the exponential map exp : ⊥S → M is a diffeomorphism. Where, ⊥S denotes the normal bundle of S. Fact 2.3 (see [3], [8]). Let M be a Riemannian manifold and G be a connected subgroup of Iso(M), e→M e be the universal Riemannian covering manifold of M with the covering map κ : M and let M and deck transformation group ∆. Then, there is a connected covering Ge of G with the covering map e and π : Ge → G , such that Ge acts isometrically on M e e (1) Each δ ∈ ∆ maps G-orbits on to G-orbits. e x)) = G(x). (2) If x ∈ M and xe ∈ π −1 (x) then κ (G(e e −1 G G e = κ (M ). (3) M e δ ge = geδ ). (4) The deck transformation group centralizes Ge (i.e., for each δ ∈ ∆ and ge ∈ G, e e G is a one point set then M = M. e (6) If M Fact 2.4. If G is a closed and connected subgroup of the isometries of a Riemannian manifold M, then the set of the fixed points of the action of G on M, M G = {x ∈ M : G(x) = x}, is a totally geodesic submanifold of M. Lemma 2.5 ([9]). If M is a connected and complete cohomogeneity k Riemannian G-manifold then k > dimM G . e is its universal covering Fact 2.6. ([4]) If M is a Riemannian manifold of negative curvature, M

83


e ∆(γ ) = γ , then ∆ is manifold and ∆ is the deck transformation group such that for a geodesic γ in M, isomorphic to (Z, +). Fact 2.7. A vector bundle over circle S1 is diffeomorphic to a cylinder over S1 or it is diffeomorphic to a cylinder over the mobious band B. 3. Results Definition 3.1. We say that a Riemannian manifold M is universally and negatively decomposable e decomposes as M e=M e1 × M e2 × ... × M ek such that for (UND), when its universal covering manifold M e ei ). each i, Mi has negative curvature, and each δ ∈ ∆ decomposes as δ = δ1 × δ2 × ... × δk , δi ∈ ISO(M Example 3.2. If M is a direct product of negatively curved manifolds then M is a UND-manifold. e have negative curvatures and ∆ is a Example 3.3. If the factors of de Rham decomposition of M e subset of the connected component of ISO(M) then M is a UND-manifold (see [9] vol. 1, page 240). e be its universal covering, by the Lemma 3.4. Let M be a UND-Riemannian manifold and let M e and an element δ in the center of ∆ such that e → M . If there is a geodesic γ in M covering map k : M 1 δ (γ ) = γ . Then M is a vector bundle over a circle S . e=M f1 × M f2 × ... × M fl , ∆ = ∆1 × ∆2 × .... × ∆l , δ = δ1 × δ2 × .... × δl , γ = Proof: Suppose that M fi . Since δ belongs to the center of γ1 × γ2 .... × γl such that for each i, δi ∈ ∆i and γi is a geodesic in M fi → R defined ∆, for each i, δi commutes with the elements of ∆i . Consider the convex functions efi : M 2 e by fi (a) = d (a, δi a) and put e → R, f (x) = ∑ fi (xi ), x = (x1 , ...,xl ). fe : M i

e Now put fe is a convex function on M. f : M → R, f (x) = fe(κ −1 (x)) f is well defined because if y, z ∈ κ −1 (x) then there is a σ in ∆ such that σ (y) = z, so fe(z) = fe(σ y) = ∑ fi (σi yi ) = ∑ d 2 (σi yi , δi σi yi ) = i

∑ d 2(σiyi, σiδiyi) = ∑ d 2(yi, δiyi) = fe(y) Put λ = κ ◦ γ . Since by assumptions, δ (γ ) = γ , then for all i, δi γi = γi . By Fact 2.3, the image of γi fi → R. Then, the image of γ is the minimum point set is the minimum point set of the function efi : M e of f , and the image of λ must be the minimum point set of f . Now, by Fact 2.2(4), the image of λ is totally convex, so it is simply closed geodesic in M and diffeomorphic to S1 . Then, by Fact 2.2(5), M is a vector bundle over S1 .

84


Theorem 3.5. Let M n+2 be a nonsimply connected UND-Riemannian manifold which is of cohomogeneity two under the action of G a closed and connected subgroup of isometries such that M G ̸= 0. / Then (a) M G is diffeomorphic to S1 . (b) M is diffeomorphic to S1 × Rn+1 or B × Rn , where B is the mobious band. e Ge is a one point set then by Fact 2.3(6), M must be Proof: Keeping the symbols of Fact 2.3, If M e Ge ≥ 2 then by Lemma 2.5, the simply connected, which is in contrast with the assumptions. If dimM e Ge = 1, and M e Ge is cohomogeneity of the action of G on M must be bigger than 3. Therefore, dimM equal to the image of a geodesic γ . By Fact 2.3(4), the elements of ∆ and Ge are commutative. So, for e x ∈ γ and δ ∈ ∆, we have each g ∈ G, gδ (x) = δ g(x) = δ (x) ⇒ δ (x) ∈ γ . Thus, ∆ is a discrete subgroup of the isometries such that ∆(γ ) = γ and by Fact 2.6, it must be isomorphic to Z. Now, by Lemma 3.4, M is a vector bundle over a circle S1 and by Fact 2.7, it is e Ge is diffeomorphic to R (the image of γ ), ∆ is isomordiffeomorphic to S1 × Rn+1 or B × Rn . Also, M phic to π1 (M) = Z and ∆(γ ) = γ . Thus M G is diffeomorphic to RZ = S1 . Example 3.6. Consider a geodesic γ in H n+1 , n ≥ 2. Let G1 be the group of rotations about γ , G2 the group of all transvections along γ and ∆ be a discrete subgroup of G2 which must be isomorn+1 phic to Z. The manifold M = H ∆ is diffeomorphic to S1 × Rn , it is of cohomogeneity two nuder the action of G = G1 and M G = S1 .

References [1] Abedi H., Alekssevsky D. V., Kashani S. M. B., Cohomogeneity one Riemannian manifolds of non-positive curvature, Differential Geometry and its Applications, 25(2007) 561-581. [2] Bishop R. L.O’Neill B., Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 ( 1969 ) 1-49. [3] Bredon. G. E., Introduction to compact transformation groups, Acad. Press, New york, London 1972. [4] Do Carmo M. P., Riemannian geometry, Brikhauser, Boston, Basel, Berlin, 1992. [5] Eberlin P. and O’Neil B., Visibility manifolds , Pasific J. Math. 46 (1973) 45-109. [6] Eberlein P., Geodesic flows in manifolds of nonpositive curvature, //math.unc.edu/Faculty/pbe/AMS− Summer.pd f .

htt p :

[7] Kobayashi S., Homogeneous Riemannian manifolds of negative curvature, Toho. Math. J. 14, 413-415(1962).

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[8] Michor P.W., Isomrtric actions of Lie groups and invariants , Lecture course at the university of Vienna, 1996/97, htt p : //www.mat.univie.ac.at/∼ michor/tgbook.ps. [9] Mirzaie R., On negatively curved G-manifolds of low cohomogeneity, Hokkaido Mathernatical Journal Vol. 38, 797-803(2009). [10] Mirzaie R., On Riemannian manifolds of constant negative curvature, J. Korean Math. Soc. 48, No. 1, 23-31(2011). [11] Mirzaie R., On homoheneous submanifolds of negatively curved Riemannian manifolds, Publ. Math. Debrecen 82 (2013) 267-275. [12] Mirzaie R. and Kashani S. M. B., On cohomogeneity one flat Riemannian manifolds, Glasgow Math. J. 44 (2002) 185190. [13] Mirzaie, R. and Kashani S. M. B., Topological properties of some cohomogeneity one Riemannian manifolds of non-positive curvature, Bull. Korean Math. Soc. 37 (2000), No. 3, pp. 587-599. [14] O’Neill B., Semi Riemannian geomerty with applications to Relativity, Academic press, New york, Berkeley 1983. [15] Podesta F. and Spiro A., Some topological propetrties of cohomogeneity one Riemannian manifolds with negative curvature, Ann. Global Anal. Geom. 14 69-79(1996). [16] Wolf J. A., Homogeneity and bounded isometries in manifolds of negative curvature, Illinos J. Math. 8 (1964) 14-18.

86

M. Faghfouri and R. Hosseinoghli :: More on Warped product Finsler manifolds

Paper No. 1.16

More on Warped Product Finsler Manifolds Morteza Faghfouri1 and Rahim Hosseinoghli 2 e-mail:[email protected], University of Tabriz, Tabriz, Iran. 2 e-mail:r [email protected], University of Tabriz, Tabriz, Iran. 1

Abstract In this paper we prove that, If (M,F) is a 2-dimensional Finsler manifold and f is a nonconstant ∂ gi j ∂ f smooth function on M satisfying k i = 0, then M is a Riemannian manifold. ∂y ∂x Keywords: Finsler manifold, Berwaldian metric, warped product metric.

1 Introduction : In [4] E. Peyghan, A. Tayebi and B. Najafi proved that a proper W P-Finsler manifold, is Berwaldian if and only if M2 is Riemannian, M1 is Berwaldian and i j ∂ f1 ∂ xi

Ck

= 0.

(∗)

Now we can ask this question: Is there any nonconstant smooth function on Finsler manifold M, which satisfies (∗)? In this paper we show that, if M is a 2-dimensional Finsler manifold and the equality (∗) holds for a nonconstant function f , then M is a Riemannian manifold. First we introduce some notions and preliminaries. S Let M be a n-dimensional C∞ manifold. Denote by Tx M the tangent space at x ∈ M, by T M := x∈M Tx M the tangent bundle of M, and by T M 0 = T M − {0} the slit tangent bundle on M. A Finsler metric on M is a function F : T M → [0,∞) which has the following properties: (i) F is C∞ on T M0 ; (ii) F is positively 1-homogeneous on the fibers of tangent bundle T M; (iii) for each y ∈ Tx M , the following quadratic form gy on Tx M is positive definite where gy (u,v) =

1 ∂2 2 [F (y + su +tv)]|s,t=0 . 2 ∂t∂ s

87


Let (M,F) be a Finsler manifold. The second and third order derivatives of 12 Fx2 := 21 F 2 (x,y) at y ∈ Tx M 0 are the symmetric forms gy and Cy on Tx M , which called the fundamental tensor and Cartan torsion, respectively. in other notation, Cy : Tx M × Tx M × Tx M → R 1d [gy+tw (u,v)]|t=0 , u,v,w ∈ Tx M 2 dt the family C := {Cy }y∈T M0 is called the cartan torsion, it is well known that C = 0 if and only if F is Riemannian. let bi be a local frame for T M, and gi j := gy (bi ,b j ), Ci jk := Cy (bi ,b j ,bk ). then gi j (x,y) = 1 ∂ 2 F 2 (x,y) 3 2 ∂g and Ci jk = 12 ∂ yikj = 14 ∂ y∂i ∂ yFj ∂ yk . For a Finsler manifold (M,F), a global vector filed G is ini j 2 ∂y ∂y ∂ ∂ duced by F on T M 0 , which in a standard coordinate (xi ,yi ) for T M 0 is given by G = yi i −2Ci (x,y) i , ∂x ∂y where 1 y ∈ Tx M Gi = gil {[F 2 ]xk yl yk − [F 2 ]xl } 4 The G is called the spry associated to (M,F). A Finsler metric F is called a Berwald metric if Gi = 1 i j k 0 2 Γ jk (x)y y is quadratic in y ∈ Tx M for any x ∈ M [2]. For a tangent vector y ∈ Tx M , define By : Tx M × Tx M × Tx M → Tx M, Ey : Tx M × Tx M → R and Dy : Tx M × Tx M × Tx M → Tx M by By (u,v,w) := ∂ ∂ Bijkl (y)u j vk wl i |x , Ey (u,v) := E jk (y)u j vk and Dy (u,v,w) := Dijkl (y)ui vk wl i |x where ∂x ∂x Cy (u,v,w) :=

Bijkl := Dijkl := Bijkl −

1 ∂ 3 Gi , E jk = Bmjkm j k l 2 ∂y ∂y ∂y

∂ E jk 2 {E jk δli + E jl δki + Ekl δ ji + l yi }. n+1 ∂y

B, E and D are called the Berwald curvature, mean Berwald curvature and Douglas curvature, respectively. Then F is called a Berwald metric, weakly Berwald metric and a Douglas metric if B = 0, E = 0 and D = 0, respectively. The notion of warped product manifold was introduced in [1] where it served to give new examples of Riemannian manifolds. On the other hand, Finsler geometry is just Riemannian geometry without the quadratic restriction. Thus it is natural to extending the construction of warped product manifolds for Finsler geometry[3]. Let (M1 ,F1 ) and (M2 ,F2 ) be two Finsler manifolds and fi : Mi → R+ ,i = 1,2 are smooth functions. Let πi : M1 × M2 → Mi ,i = 1,2 be the natural projection maps. The product manifold M1 × M2 endowed with the metric F : T M10 × T M20 → R given by q F(y,v) = f22 (π2 (y))F12 (y) + f12 (π1 (y))F22 (v) is considered, where T M10 = T M1 − {0} and T M20 = T M2 − {0} . The metric defined above is a Finsler metric. The product manifold M1 × M2 with the metric F(y) = F(y,v) for (y,v) ∈ T M10 × T M20 defined

88


above will be called the doubly warped product (DWP) of the manifolds M1 and M2 and fi ,i = 1,2 will be called the warping function. We denote this warped by M1 f2 × f1 M2 .If f2 = 1 , then we have a waperd product manifold. If fi ,i = 1,2 is not constant, then we have a proper DW P-manifold. Let (M1 ,F1 ) and (M2 ,F2 ) be two Finsler manifolds. Then the functions gi j (x,y) =

1 ∂ 2 F12 (x,y) 2 ∂ yi ∂ y j

gαβ (u,v) =

1 ∂ 2 F22 (u,v) 2 ∂ vα ∂ vβ

define a Finsler tensor field of type (0,2) on T M10 and T M20 , respectively. Now let M1 × f1 M2 be a warped Finsler manifold and let x ∈ M and y ∈ Tx M, where x = (x,u),y = (y,v), M = M1 × M2 and Tx M = Tx M1 ⊕ Tu M2 . Then we conclude that 2 2 1 ∂ F (x,u,y,v) gi j 0 gab (x,u,y,v) = = 0 f12 gαβ 2 ∂ ya ∂ yb where ya = (yi ,vα ), yb = (y j ,vβ ) and gi j = gi j , gab = f12 gαβ , giβ = gα j = 0 and i, j,... ∈ {1,2,...,n1 }α,β ,... ∈ {1,2,...,n2 },a,b,... ∈ {1,2,...n1 ,n1 + 1,...,n1 + n2 }, in this case dim(M1 ) = n1

dim(M2 ) = n2

dim(M1 × M2 ) = n1 + n2 .

so the spray coeffcients of warped product are given by 1 ∂ f2 Gi (x,u,y,v) = Gi (x,y) − gih 1h F22 4 ∂x

Gα (x,u,y,v) = Gα (u,v) +

1 αλ ∂ f12 ∂ F22 l g y ∂ xl ∂ vλ 4 f12

the Berwald curvature of (M1 × f M2 ) is as follows: 1 ∂ 3 gkh ∂ f12 2 F , 4 ∂ yi ∂ y j ∂ yl ∂ xh 2 1 ∂ 2 gkh ∂ f12 ∂ F22 , Bkiβ l = − 4 ∂ yl ∂ yi ∂ xh ∂ vβ ∂ f 2 ∂ gkh Bkαβ l = − 1h l gαβ , ∂x ∂y ∂ f2 Bkαβ λ = − 1h gkhCαβ λ , ∂x Bkijl = Bkijl −

γ

γ

Bαβ λ = Bαβ λ , γ

Biβ λ = 0 γ

Bi jλ = 0, γ

Bi jk = 0.

Proposition 1.1 ([4]). Let (M1 × f1 M2 ,F) be a proper W P-Finsler manifold. Then (M1 × f1 M2 ,F) is Berwaldian if and only if M2 is Riemannian, M1 is Berwaldian and i j ∂ f1 ∂ xi

Ck

= −2

∂ gi j ∂ f1 = 0. ∂ yk ∂ xi

89


Theorem 1.2 ([4]). Let (M1 f2 × f1 M2 ,F) be a proper DW P-Finsler manifold. Then (M1 f2 × f1 M2 ,F) is weakly Berwald if and only if M2 and M1 are weakly Berwalds and i j ∂ f1 ∂ xi

Ck

γν ∂ f2 ∂ uν

= Cγ

= 0.

Proposition 1.3 ([4]). Let (M1 × f1 M2 ,F) be a proper W P-Finsler manifold. Then (M1 × f1 M2 ,F) is Douglas if and only if M2 is Riemannian, M1 is Berwaldian and i j ∂ f1 ∂ xi

Ck

= 0.

2 Main result Theorem 2.1. If (M,F) is a 2-dimensional Finsler manifold and f is nonconstant smooth function on M satisfying ∂ gi j ∂ f = 0, ∂ yk ∂ xi

(2.1)

then M is a Riemannian manifold. Proof. Let f be a nonconstant smooth function on M which satisfies (2.1). then we have gi j

∂f = ci (x), ∂xj

where ci (x) is a smooth function on M. The above Equ. (2.2) implies that

(2.2) ∂f = ci (x)gik . and ∂ xk

∂ f 1 i ∂ 2F 2 = c (x) i j ∂xj 2 ∂y ∂y

(2.3)

By integrating (2.3) with respect to y j we obtain 2 By Choosing u = F 2 ,B j =

∂ f j i ∂ F2 y = c (x) i ∂xj ∂y

(2.4)

∂f in the equation (2.4), we have the following PDE equation ∂xj c1

∂u 2 ∂u + c 2 = 2B1 y1 + 2B2 y2 . ∂ y1 ∂y

90

(2.5)


The general solution of PDE equation (2.5) are given by F2 = u =

B1 1 2 B2 2 2 (y ) + 2 (y ) + ϕ(c2 y1 − c1 y2 ) c1 c

(2.6)

where ϕ is an arbitrary smooth one variable function [5]. Since u is homogeneous of degree 2 so ϕ is homogeneous of degree 2, too. That ϕ is one variable implies that ϕ(t) = At 2 , where A is a constant, so we have F2 =

B1 1 2 B2 2 2 (y ) + 2 (y ) + A(c2 y1 − c1 y2 )2 . c1 c

The Cartan torsion Ci jk =

(2.7)

1 ∂ gi j 1 ∂ 3 F 2 = =0 2 ∂ yk 4 ∂ yi ∂ y j ∂ yk

thus M is Riemannian. Corollary 2.2. Let (M1 ,F1 ) and (M2 ,F2 ) be Finsler manifolds with dimM1 = 2, dimM2 = n2 and fi : Mi → R,i = 1,2 are positive smooth functions. 1. A proper (2 + n2 )-dimensional W P-Finsler manifold M1 × f1 M2 is a Berwald manifold, if and only if it is a Riemannian manifold. 2. A proper (2 + 2)-dimensional DW P-Finsler manifold M1 f2 × f1 M2 is a weakly Berwald manifold, if and only if it is Riemannian manifold (dimM2 = 2). 3. A proper (2 + n2 )-dimensional W P-Finsler manifold M1 × f1 M2 is a Douglas manifold, if and only if it is a Riemannian manifold.

References [1] R. L. B ISHOP, AND B. O’N EILL , Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 1–49. [2] S.-S. C HERN , AND Z. S HEN , Riemann-Finsler geometry, vol. 6 of Nankai Tracts in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. [3] L. KOZMA , R. P ETER , AND C. VARGA , Warped product of Finsler manifolds. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 44 (2001), 157–170 (2002). [4] E. P EYGHAN , A. TAYEBI , AND B. NAJAFI , Doubly warped product Finsler manifolds with some non-Riemannian curvature properties. Ann. Polon. Math. 105, 3 (2012), 293–311. [5]

S NEDDON , Elements of partial differential equations. Dover Publications Inc., Mineola, NY, 2006. Unabridged republication of the 1957 original. V

91

M. Nadjafikah, A. Mahdavi and M. Toomanian :: Two approaches to the calculation of approximate symmetry of a Ostrovsky equation with small parameter

Paper No. 1.17

Two approaches to the calculation of approximate symmetry of a Ostrovsky equation with small parameter Mehdi Nadja…kah

Abolhassan Mahdaviy

Megerdich Toomanianz

m_nadja…[email protected]

[email protected]

[email protected]

Abstract In this paper, two method of approximate symmetries for partial di¤erential equations with a small parameter is applied to a perturbed nonlinear Ostrousky equation. In order to compute the …rstorder approximate symmetry, we have applied two methods which one of them was proposed by Baikov et al. in which the in…nitesimal generator is expanded in a perturbation series; whereas the other method that we have used, due by Fushchich and Shtelen [3] is based on the expansion of the dependent variables in perturbation series. Especially, an optimal system of one dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.

Keywords: approximate symmetry, approximate solution, perturbed Ostrovsky equation

1

Introduction

Nonlinear problems arise in many …elds of science and engineering.Lie group theory greatly simpli…es many nonlinear partial di¤erential equations. Many PDEs in application depend on small parameter, so it is of great importance and interest to …nd approximate solutions One of the techniques widely applied in analyzing nonlinear problems is the perturbation analysis. Perturbation theory comprises mathematical methods that are applied to obtain an approximate solution to a problem which can not be solved exactly. Indeed, this procedure is performed by expanding the dependent variables asymptotically in terms of a small parameter. In order to combine the power of the Lie group theory and perturbation analysis, two di¤erent approximate symmetry theories have been developed recently. The …rst method is due to Baikov, Gazizov and Ibragimov [1, 2]. Successively another method for obtaining approximate symmetries was introduced by Fushchich and Shtelen [5].

92

M. Nadjafikah, A. Mahdavi and M. Toomanian :: Two approaches to the calculation of approximate symmetry of a Ostrovsky equation with small parameter

In the method proposed by Baikov, Gazizov and Ibragimov, the Lie operator is expanded in a perturbation series other than perturbation for dependent variables as in the usual case. In other words, assume that the perturbed di¤erential equation be in the form: F (z) = F0 (z) + "F1 (z), where z = (x; u; u(1) ;

; u(n) ), F0 is the unperturbed equation, F1 (z) is the perturbed term and X = X0 +"X1

is the corresponding in…nitesimal generator. The exact symmetry of the unperturbed equation F0 (z) is denoted by X0 and can be obtained as X0 F0 (z) H=

F0 (z)=0

= 0. Then, by applying the auxiliary function

1h X0 (F0 (z) + "F1 (z)) "

F0 +"F1 =0

i

(1)

vector …eld X1 will be deduced from the following relation: X1 F0 (z)

F0 =0

+ H = 0:

(2)

.Finally, after obtaining the approximate symmetries, the corresponding approximate solutions will be obtained via the classical Lie symmetry method [6]. In the second method due to Fushchich and Shtelen, …rst of all the dependent variables are expanded in a perturbation series. In the next step, terms are then separated at each order of approximation and as a consequence a system of equations to be solved in a hierarchy is determined. Finally, the approximate symmetries of the original equation is de…ned to be the exact symmetries of the system of equations resulted from perturbations [4, 5, 12]. In this paper, we will apply two methods in order to present a comprehensive analysis of the approximate symmetries of perturbed Ostrovsky equation utx + u2x + uuxx + u = uxx : where 0