On Randers-Conformal Change of Finsler Space ... - Semantic Scholar

International Journal of Contemporary Mathematical Sciences Vol. 11, 2016, no. 9, 415 - 423 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2016.6846

On Randers-Conformal Change of Finsler Space with Special (α, β)-Metrics Sruthy Asha Baby Department of Mathematics and Statistics Banasthali University Banasthali, Rajasthan-304022, India Gauree Shanker Centre for Mathematics and Statistics Central University of Punjab, Bathinda Bathinda, Punjab-151001, India c 2016 Sruthy Asha Baby and Gauree Shanker. This article is distributed Copyright under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In the present paper, we first find the fundamental metrics tensor of the Randers conformally transformed special (α, β)-metrics F = α + 2 β + k βα ( and k 6= 0 are constants) and then we find a condition under which the Randers conformally transformed special (α, β)-metrics is locally dually flat.

Mathematics Subject Classification: 53B40, 53C60 Keywords: Dually flat, Projective flatness, Finsler space with (α, β)metrics, Conformal transformation

1

Introduction q

Let α = aij y i y j be a Riemannian metrics and β = bi y i be a differential 1form of an n-dimensional differentiable manifold M n . The Finsler metrics F

416

Sruthy Asha Baby and Gauree Shanker

= α + β was originated by G. Randers in his unified field theory and named 2 the Randers space by R. S. Ingarden [8]. The Kropina metrics F = αβ , a kind of (α, β)-metrics was investigated by C. Shibata [13] and treated by Ingarden in his paper concerned with thermodynamics [9]. In 1976, Hashiguchi [7] introduced the conformal change of Finsler metrics given by F¯ = eσ(x) F . In particular, he studied the special conformal transformation called as C-conformal. This transformation has also been studied by many authors ([10], [15]). In 1984, Shibata [12] extended the notion of β-change to a general case in Finsler geometry, i.e., F¯ = f (F, β). Under this transformation, he also studied the change of torsion tensor and curvature tensor, and also dealt with some special Finsler spaces corresponding to specific forms of this function. In 2008, Abed ([1], [2]) generalized the conformal, Randers and generalized Randers changes by a new transformation F¯ = eσ(x) F + β. In addition to this, he also obtained the correlation between some relevant tensors associated with (M, F ) and the corresponding tensors associated with (M, F¯ ). Further, he discussed the invariant and σ-invariant properties and determined an association between the Cartan connection associated with (M, F¯ ) and the transformed Cartan connection associated with (M, F¯ ). The main purpose of thecurrent paper is to investigate the special (α, β) 2 σ ¯ metrics F = e α + β + k βα + β, which is obtained by the Randers conformal 2

change of the special (α, β)- metrics F = α + β + k βα ( and k 6= 0). The paper is organized as follows: Starting with literature survey in section one, we find fundamental metrics tensor g¯ij and its inverse g¯ij in section two [see Theorem 2.1 and 2.2 ] for the Randers conformally transformed metrics. In section three, first we find the spray coefficients for the transformed metrics (see lemma 3 ) and then we find the necessary and sufficient conditions for this metrics to be locally dually flat (see Theorem 3.1 ).

2

Fundamental metrics tensor of Randers conformal change of special (α, β) metrics q

A Finsler metrics (1) where α = aij y i y j is a Riemannian metrics and β = bi (x)y i is a differential one form, which was introduced by physicst Randers in 1941 from view point of general theory of relativity. Further, Matsumoto, Ingarden ([5] studied this metrics as a Finsler metrics and investigated its properties. In 2008, S. Abed introduced a Finsler metrics F¯ = eσ(x) F + β, where F is a Finsler metrics, σ(x) is a conformal factor and β is a one form. This metrics is called Randers conformal change of Finsler metrics.

On Randers-conformal change of Finsler space with special (α, β)-metrics

417

Here, we consider the Randers conformal change of the special (α, β)-metrics 2 F = α + β + k βα given by 2

β F¯ = eσ α + (eσ + 1)β + keσ . (1) α The differentiation of (1) with respect to y i yields the normalized supporting element li given by l¯i = eσ li + bi , In view of (2), we have 2b eσ yi βyi i l¯i = + (eσ + 1)bi + eσ kβ − 3 , (2) α α α Differentiation of (2) with respect to y j yields 1 σ β2 1 3β 2 2eσ k σ bi bj e − 2 aij + 3 − e y y + i j α α α α2 α 2eσ kβ 2β − 3 yj bi − 3 yi bj , α α From (2) and (3), we obtain ¯ ij = F¯ ¯lij , h = p1 aij + p2 yi yj + p3 bi bj + p4 (yj bi + yi bj ),

¯lij =

(3)

(4)

where 1 σ β 2 σ eσ kβ 2 σ p1 = e − 2 e α + (e + 1)β + , α α α eσ kβ 2 1 3β 2 σ σ σ − e e α + (e + 1)β + , (5) p2 = α3 α2 α h β kβ 2 i p3 = 2eσ k eσ + (eσ + 1) + eσ 2 , α α 2eσ kβ σ (eσ + 1)β eσ kβ 2 p4 = − 2 e + + . α α α2 From (2) and (4), the fundamental metrics tensor g¯ij of Finsler space (M, F¯ ) is given by: ¯ ij + ¯li ¯lj , g¯ij = h = ρaij + ρ0 bi bj + ρ1 (bi αj + bj αi ) + ρ2 (αi αj ),

(6)

where 1 σ β 2 σ eσ kβ 2 e − 2 e α + (eσ + 1)β + , α α α 2e2σ k(3kβ 2 + α2 ) + 4eσ k(eσ + 1)αβ = , α2 −4e2σ k 2 β 3 + eσ (eσ + 1)(α2 − 3kβ 2 )α = , α5 eσ (2 − 3eσ k)β 2 − eσ (eσ + 1)βα = . α6

ρ = ρ0 ρ1 ρ2

(7)

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Theorem 2.1 For the Randers conformal transformed (α, β)- metrics F¯ = 2 eσ F + β, where F = α + β + k βα , the fundamental metrics tensor g¯ij is given by equations (6) and (7). The contravariant metrics tensor g¯ij of Finsler space (M, F¯ ) is given by g¯ij = ρ−1 {aij − τ bi bj − ηY i Y j } n δ i j µα = ρ−1 aij − bb − 2 1 + δb 1 + {α + (λ + ξ)β + λξb2 α)}µ yi y j o + λbi + λbj , α α

(8)

where 1 σ β 2 σ eσ kβ 2 σ , e − e α + (e + 1)β + α3 α2 α l1 α6 + l2 α5 β + l3 α4 β 2 + l4 α3 β 3 + l5 α2 β 4 + l6 αβ 5 − 16e4σ k 4 β 6 , δ = l7 α11 β + l8 α10 β 2 + l9 α9 β 3 + l10 α8 β 4 + l11 α7 β 5 + l12 α6 β 6 eσ (2 − 3eσ k)β 2 − eσ (eσ + 1)αβ , µ = α2 eσ α2 − β 2 eσ α2 + (eσ + 1)αβ + eσ kβ 2 ρ =

λ = =

(9)

αρ1 ρ − (ρ2 ρ0 − ρ21 )β , αρρ2 + (ρ2 ρ0 − ρ21 )b2 α h

α9 q1 + α8 βq2 + α7 β 2 q3 + α6 β 3 q4 + α5 β 4 q5 + α4 β 5 q6 + α3 β 6 q7 +

α2 β 7ih q8 − α6 βq9 − α5 β 2 q10 − α4 β 3 q11 − α3 β 4 q12 − α2 β 5 q13 − αβ 6 q14 −q15 α11 β 2 q10 + α10 β 3 q20 + α9 β 4 q30 + α8 β 5 q40 + α7 β 6 q50 + b2 (α7 q9 + i

α6 βq10 + α5 β 2 q11 + α4 β 3 q12 + α3 β 4 q13 + α2 β 5 q14 + αβ 5 q15 ) , −4eσ k 2 αβ 3 + (eσ + 1)α2 (α2 − 3kβ 2 ) , ξ = −(eσ + 1)αβ + (2 − 3eσ k)β 2 in which, l1 l2 l3 l4 l5 l6 l7 l8 l9

= = = = = = = = =

e2σ (eσ + 1)2 , −2e3σ k(eσ + 1), 3e3σ k 2 (4 − 6) − 9k 2 (1 + 2k)e4σ − 9k 2 e2σ , (eσ + 1)(2e3σ k(2k − 3) + 8e2σ k), 6k 2 e3σ (2 − 3eσ k) − 9k 2 e2σ (eσ + 1)2 , 24k 3 e3σ (eσ + 1), −e3σ (eσ + 1), 2e3σ (1 − ) − e4σ (3k + 2 ) − e2σ , (eσ + 1)(3e2σ − 4e3σ k),


l10 l11 l12 q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q10 q20 q30 q40 q50

= = = = = = = = = = = = = = = = = = = = = = =

419

(5k + 2 )e3σ + 2( − 1)e2σ − 3e4σ k 2 + eσ , (eσ + 1)(4e2σ k − 2eσ ), e2σ k(2 − 3eσ k), e3σ (eσ + 1), e2σ (eσ + 1)2 , −e2σ (eσ + 1)(1 + 2eσ k), −e4σ (4k + 22 )k + e3σ (2 − 6k) + e2σ (2 − 3k) + eσ , (2 − 7eσ k)(eσ + 1)e2σ k, −4e4σ k + e3σ (32 k + 4) + 6ke2σ , e2σ (eσ + 1)(4 + 3k 2 ), 4e3σ k 3 , −2ke4σ (k − 52 ) + 4e3σ k(1 − 5) − 6ke2σ − 4eσ k, −(eσ + 1){(2k − 3)2e3σ k + 8e2σ k}, e2σ (eσ + 1)2 , −2e3σ k(eσ + 1), −(2 + 2k)9e4σ k 2 + (2 − 3)6k 2 e3σ + 9k 2 e2σ , 24e3σ (eσ + 1)k 3 , −16e4σ k 4 β 7 . 2e3σ (1 − ) − e4σ (3k + 2 ) − e2σ , (eσ + 1)(3e2σ − 4ke3σ ), −e3σ (eσ + 1), e2σ (keσ − 1)(2 − 3eσ k) + eσ (eσ + 1)k, e2σ k(2 − 3eσ k).

Theorem 2.2 For the Randers conformal transformed (α, β)- metrics F¯ = 2 eσ F + β, where F = α + β + k βα , the contravariant metrics tensor g¯ij of F¯ is given by equations (8) and (9).

3

Locally dually flatness of transformed (α, β)metrics.

The notion of dually flat Riemannian metrics was introduced by S. I. Amari and H. Nagaoka ([3],[4]), when they studied the information geometry on Riemannian manifolds. In Finsler geometry, Shen extended the notion of locally dually flatness for Finsler metrics. Dually flat Finsler metrics form a special and valuable class of Finsler metrics in Finsler information geometry, which plays a very important role in studying flat Finsler information structure.

420


Information geometry has been emerged from investigating the geometricsal structure of the family of probability distributions. A Finsler space F n = (M, F(x,y)) is called the (α, β)-metrics, if there exists a 2-homogenous function L of two variables such that the Finsler metrics F: TM −→ < is given by: F 2 (x, y) = L(α(x, y), β(x, y)),

(10)

where α2 (x, y) = aij (x)y i y j , α is Riemannian metrics on M, and β(x, y) = bi (x)y i is a 1-form on M. A Finsler metrics F = F (x, y) on a manifold M n is said to be locally dually flat, if at any point there is a standard coordinate system (xi , y i ) in TM such that [F 2 ]xk yl y k = 2[F 2 ]xl .

(11)

In this case, the coordinate (xi ) is called an adapted local coordinate system. Every locally Minkowskian metrics is locally dually flat. Consider the Randers conformally transformed metrics F¯ = eσ(x) F + β, where F is a special (α, β)-metrics. If φ(s) = eσ (1 + s + ks2 ) + s; s = αβ , then (α, β)-metrics is exactly the metrics of the form of Randers conformally transformed special (α, β)-metrics. ¯ i (x, y) and Giα (x, y) denote the spray coefficients of F¯ and α respectively. Let G ¯ i of F¯ in terms of α and β, we To express formulae for the spray coefficients G need to introduce some notations. Let bi:j be a covariant derivative of bi with respect to y j . Denote 1 1 sij = (bi|j − bj|i ), rij = (bi|j + bj|i ), 2 2 i ih i i sj = a shj , sj = bi sj = sij b , rj = rij bi , r0 = rj y j , s0 = sj y j , r00 = rij y i y j . ¯ i of the Randers conformally transformed special The spray coefficients G (α, β)- metrics are related to Giα by i

¯ i = Giα + αQsi0 + Θ(−2αQs0 + r00 ) y + ψ(−2αQs0 + r00 )bi , G α where Q

=

φ0 eσ ( + 2ks) + 1 = , φ − sφ0 eσ (1 − ks2 )

Θ

=

(φ − sφ0 )φ0 2φ[(φ − sφ0 ) + (b2 − s2 )φ00 ]

(12)


= ψ

= =

−[4k2 eσ (1

421

2e2σ k − (e2σ 2 + 1)s + (2e2σ k2 − 6ke2σ − 6eσ k)s2 − (6k2 e2σ )s3 , − [2k(eσ + 1)(eσ + 1)]s3 − [4e2σ k(1 − b2 k)]s2 + [2eσ (eσ + 1)]s + 2eσ (1 + 2b2 k)

eσ )]s4

+ φ” , 2{(φ − sφ0 ) + (b2 − s2 )φ00 } eσk . 1 + k(2b2 − 3s2 )

Here bi = aij bj , and b2 = aij bi bj = bj bj .

From (11), we have the following

Theorem 3.1 An (α, β)-metrics F¯ = αφ(s) = α(eσ + {eσ + 1}s + eσ ks2 ), where s = αβ , is dually flat on an open subset U ⊂ Rn if and only if 2 σ 2 3 2{α5 e2σ + eσ (eσ + 1)α4 β + eσ (eσ + 1)kα2 β 3 − e2σ k 2 αβ 4 }α2 aml Gm α + [2k e α β

+{2keσ ( + k) + k2 e2σ + k}α3 β 2 + α4 β{2eσ + 2eσ k + 2 e2σ + 1} + eσ (eσ + 1)α5 ] h ∂Gm α (3sl0 − rl0 )α3 − α3 (eσ α2 + {eσ + 1}αβ + eσ kβ 2 ) eσ (α2 − kβ 2 )ym + {2keσ βα ∂y l i h ∂Gm α m +(eσ + 1)α2 }αbm α(r + 2b G )y + {4k 2 eσ β 3 + 2(2keσ + 2keσ k + k2 e2σ 00 m l α ∂y l 2σ 2 3 +k)αβ 2 + α2 β(4eσ + 4eσ k + 22 e2σ + 2) + β 3 2eσ (eσ + 1)}ym Gm α + [e (2k + )α +(2eσ + 1)α3 + {6keσ (eσ + 1)}α3 β + 6e2σ k 2 β 2 α]{αr00 + 2(α2 bm − βym )Gm α}

i

(α2 bl − βyl ) = 0.

By direct computation, F¯ is locally dually flat on U if and only if [F¯ 2 ]xk yl y k = 2[F¯ 2 ]xl which implies αφ2 (αxk yl y k − 2αxl ) + α2 φφ0 (sxk yl y k − 2sxl ) + 2αφφ0 (αyl sxk y k +syl αxk y k ) + α2 (φ02 + φφ00 )(sxk y k )syl = 0, On the other hand, α xl =

1 ∂Gm 2 yl α ym , αxk y k = Gm , α ym , α l = l α ∂y α α

1 1 ∂Gm αbl − syl bm;l y m + 2 (αbm − sym ) αl , syl = , α α ∂y α2 r00 2 s xk y k = + 2 (αbm − sym )Gm α, α α 2 1 ∂Gm α αxk yl y k − 2αxl = 3 (aml α2 − ym yl )Gm − ym , α α α ∂y l

s xl =

(13) (14) (15) (16)

r00 2 4yl 2 yl m y + s − (αb − sy )G + bm l l0 m m α α3 α α4 α2 α αbi − syl 1 1 ∂Gm α m m − y − sa G − b y − (αb − sy ) . (17) m ml m;l m m α α2 α α2 ∂y l sxk yl y k − 2sxl = −

422


Putting (13), (14), (15), (16) and (17) into (13) and noting bm;l y m = r0l + s0l we get h

0 3 2 0 2φ(φ − sφ0 )α2 aml Gm α + φφ (3sl0 − rl0 )α − α φ (φ − sφ )ym

∂Gm α ∂y l

i ∂Gm α 0 m 02 00 + φφ0 α(r00 + 2bm Gm α )yl + [2φφ ym Gα + (φ + φφ ) ∂y l (αr00 + 2(αbm − sym )Gm α )](αbl − sym ) = 0

+αφ0 bm

which implies 2 σ 3 2{e2σ + eσ (eσ + 1)s + eσ (eσ + 1)ks3 − e2σ k 2 s4 }α2 aml Gm α + [2k e s + {2keσ ( + k) + k2 e2σ + k}s2 + s{2eσ + 2eσ k + 2 e2σ + 1} + eσ (eσ + 1)] h ∂Gm (3sl0 − rl0 )α3 − α2 (eσ + {eσ + 1}s + eσ ks2 ) eσ (1 − ks2 )ym αl + {2keσ s + ∂y mi h ∂G 2 σ 3 σ σ (eσ + 1)}αbm αl α(r00 + 2bm Gm α )yl + {4k e s + 2(2ke + 2ke k + ∂y k2 e2σ + k)s2 + s(4eσ + 4eσ k + 22 e2σ + 2) + 2eσ (eσ + 1)}ym Gm α + 2σ 2 σ σ σ 2σ 2 2 [e (2k + ) + 2e + 1 + {6ke (e + 1)}s + 6e k s ]{αr00 + 2(αbm −

i

sym )Gm α } (αbl − syl ) = 0. This completes the proof.

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