Mar 17, 2014 - The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem ...
PSEUDO-RIEMANNIAN GEOMETRY, δ-INVARIANTS AND APPLICATIONS
PSEUDO-RIEMANNIAN GEOMETRY, δ-INVARIANTS AND APPLICATIONS The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudoRiemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades.
PSEUDO-RIEMANNIAN GEOMETRY, δ-INVARIANTS AND APPLICATIONS
Bang-Yen Chen
Chen
World Scientific www.worldscientific.com 8003 hc
ISBN-13 978-981-4329-63-7 ISBN-10 981-4329-63-0
World Scientific
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Contents
Preface
vii
Foreword
ix
1.
Pseudo-Riemannian Manifolds
1
1.1 1.2 1.3
2 3 5 5
1.4 1.5 1.6 1.7 1.8 1.9 1.10 2.
Symmetric bilinear forms and scalar products . . . . . . . Pseudo-Riemannian manifolds . . . . . . . . . . . . . . . . Physical interpretations of pseudo-Riemannian manifolds 1.3.1 4D spacetimes . . . . . . . . . . . . . . . . . . . . 1.3.2 Kaluza-Klein theory and pseudo-Riemannian manifolds of higher dimension . . . . . . . . . . . . . . Levi-Civita connection . . . . . . . . . . . . . . . . . . . . Parallel translation . . . . . . . . . . . . . . . . . . . . . . Riemann curvature tensor . . . . . . . . . . . . . . . . . . Sectional, Ricci and scalar curvatures . . . . . . . . . . . . Indefinite real space forms . . . . . . . . . . . . . . . . . . Lie derivative, gradient, Hessian and Laplacian . . . . . . Weyl conformal curvature tensor . . . . . . . . . . . . . .
Basics on Pseudo-Riemannian Submanifolds 2.1 2.2 2.3 2.4 2.5 2.6 2.7
Isometric immersions . . . . . . . . . . . . . . . . . . Cartan-Janet’s and Nash’s embedding theorems . . . Gauss’ formula and second fundamental form . . . . Weingarten’s formula and normal connection . . . . Shape operator of pseudo-Riemannian submanifolds Fundamental equations of Gauss, Codazzi and Ricci Fundamental theorems of submanifolds . . . . . . . . xxv
7 9 11 15 17 19 21 24 25
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26 27 28 30 33 34 38
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2.11 2.12
Robertson-Walker Spacetimes 5.1 5.2 5.3 5.4 5.5 5.6 5.7
6.
Totally geodesic submanifolds . . . . . . . . . . . . . . Parallel submanifolds of (indefinite) real space forms . Totally umbilical submanifolds . . . . . . . . . . . . . Totally umbilical submanifolds of Ssm (1) and Hsm (−1) Pseudo-umbilical submanifolds of Em s . . . . . . . . . . Pseudo-umbilical submanifolds of Ssm (1) and Hsm (−1) Minimal Lorentz surfaces in indefinite real space forms Marginally trapped surfaces and black holes . . . . . . Quasi-minimal surfaces in indefinite space forms . . .
Basics of warped products . . . . . . . . . . . . . . Curvature of warped products . . . . . . . . . . . . Warped product immersions . . . . . . . . . . . . . Twisted products . . . . . . . . . . . . . . . . . . . Double-twisted products and their characterization
44 47 52
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53 55 57 60 63 64 67 71 75 77
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78 80 83 86 89 91
Cosmology, Robertson-Walker spacetimes and Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . . 91 Basic properties of Robertson-Walker spacetimes . . . . . 94 Totally geodesic submanifolds of RW spacetimes . . . . . 98 Parallel submanifolds of RW spacetimes . . . . . . . . . . 99 Totally umbilical submanifolds of RW spacetimes . . . . . 101 Hypersurfaces of constant curvature in RW spacetimes . . 105 Realization of RW spacetimes in pseudo-Euclidean spaces 106
Hodge Theory, Elliptic Differential Operators and Jacobi’s Elliptic Functions 6.1
39 41
53
Warped Products and Twisted Products 4.1 4.2 4.3 4.4 4.5
5.
A reduction theorem of Erbacher-Magid . . . . . . . . . . Two basic formulas for submanifolds in Em . . . . . . . . s Relationship between squared mean curvature and Ricci curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship between shape operator and Ricci curvature Cartan’s structure equations . . . . . . . . . . . . . . . . .
Special Pseudo-Riemannian Submanifolds 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
4.
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2.8 2.9 2.10
3.
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Operators d, ∗ and δ . . . . . . . . . . . . . . . . . . . . . 108
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6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.
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111 112 115 117 120 124 125 127
Order and type of submanifolds . . . . . . . . . . . . . . Minimal polynomial criterion . . . . . . . . . . . . . . . A variational minimal principle . . . . . . . . . . . . . . Classification of 1-type submanifolds . . . . . . . . . . . Finite type immersions of compact homogeneous spaces Submanifolds of Em s satisfying ∆H = λH . . . . . . . . Submanifolds of H m (−1) satisfying ∆H = λH . . . . . Submanifolds of S1m (1) satisfying ∆H = λH . . . . . . . Biharmonic submanifolds . . . . . . . . . . . . . . . . . Null 2-type submanifolds . . . . . . . . . . . . . . . . . Spherical 2-type submanifolds . . . . . . . . . . . . . . . 2-type hypersurfaces in hyperbolic spaces . . . . . . . .
. . . . . . . . . . . .
Total Mean Curvature 8.1 8.2 8.3 8.4 8.5 8.6 8.7
9.
Hodge-Laplace operator . . . . . . . . . . . . . . . Elliptic differential operator . . . . . . . . . . . . . Hodge-de Rham decomposition and its applications The fundamental solution of heat equation . . . . . Spectra of some important Riemannian manifolds . Spectra of flat tori . . . . . . . . . . . . . . . . . . Heat equation, Jacobi’s elliptic and theta functions
Submanifolds of Finite Type 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12
8.
xxvii
128 131 134 137 138 140 142 144 145 148 152 156 161
3
Total mean curvature of tori in E . . . . . . . . Total mean curvature and conformal invariants . Total mean curvature for arbitrary submanifolds Total mean curvature and order of submanifolds Conformal property of λ1 vol(M ) . . . . . . . . . Total mean curvature and λ1 , λ2 . . . . . . . . . Total mean curvature and circumscribed radii . .
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162 164 167 171 175 176 178
Pseudo-K¨ ahler Manifolds
183
9.1 9.2 9.3 9.4 9.5 9.6
184 187 190 192 196 198
Pseudo-K¨ ahler manifolds . . . . . . . . . . . . . . . . . . . Pseudo-K¨ ahler submanifolds . . . . . . . . . . . . . . . . . Purely real submanifolds of pseudo-K¨ahler manifolds . . . Dependence of fundamental equations for Lorentz surfaces Totally real and Lagrangian submanifolds . . . . . . . . . CR-submanifolds of pseudo-K¨ahler manifolds . . . . . . .
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. . . . . . . .
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Pseudo-Riemannian submersions . . . . . . . . . . O’Neill integrability tensor and O’Neill’s equations Submersions with totally geodesic fibers . . . . . . Submersions with minimal fibers . . . . . . . . . . A cohomology class for Riemannian submersion . . Geometry of horizontal immersions . . . . . . . . .
Contact pseudo-Riemannian metric manifolds . . . Sasakian manifolds . . . . . . . . . . . . . . . . . . Sasakian space forms with definite metric . . . . . Sasakian space forms with indefinite metric . . . . Legendre submanifolds via canonical fibration . . . Contact slant submanifolds via canonical fibration
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228 229 230 234 237 239 241
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Motivation . . . . . . . . . . . . . . . . . . . . . . . . . Definition of δ-invariants . . . . . . . . . . . . . . . . . . δ-invariants and Einstein and conformally flat manifolds Fundamental inequalities involving δ-invariants . . . . . Ideal immersions via δ-invariants . . . . . . . . . . . . . Examples of ideal immersions . . . . . . . . . . . . . . . δ-invariants of curvature-like tensor . . . . . . . . . . . . A dimension and decomposition theorem . . . . . . . . .
Some Applications of δ-invariants
206 207 209 211 214 216 218 221 227
. . . . . .
δ-invariants, Inequalities and Ideal Immersions 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8
14.
. . . . . . . .
Contact Metric Manifolds and Submanifolds 12.1 12.2 12.3 12.4 12.5 12.6
13.
Para-K¨ ahler manifolds . . . . . . . . . . . . . . . . Para-K¨ ahler space forms . . . . . . . . . . . . . . . Invariant submanifolds of para-K¨ahler manifolds . Lagrangian submanifolds of para-K¨ahler manifolds Scalar curvature of Lagrangian submanifolds . . . Ricci curvature of Lagrangian submanifolds . . . . Lagrangian H-umbilical submanifolds . . . . . . . PR-submanifolds of para-K¨ahler manifolds . . . .
205
Pseudo-Riemannian Submersions 11.1 11.2 11.3 11.4 11.5 11.6
12.
Slant submanifolds of pseudo-K¨ahler manifolds . . . . . . 202
Para-K¨ ahler Manifolds 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8
11.
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251 252 254 260 268 270 271 275 279
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Contents
14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 15.
15.7 15.8
. . . . . . . .
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A vanishing theorem for Lagrangian immersions . . . . . Obstructions to Lagrangian isometric immersions . . . . . Improved inequalities for Lagrangian submanifolds . . . . Totally real δ-invariants δkr and their applications . . . . . Examples of strongly minimal K¨ahler submanifolds . . . . K¨ ahlerian δ-invariants δ c and their applications to K¨ahler submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . Applications of δ-invariants to real hypersurfaces . . . . . Applications of δ-invariants to para-K¨ahler manifolds . . .
δ-invariants and submanifolds of Sasakian space forms δ-invariants and Legendre submanifolds . . . . . . . . Scalar and Ricci curvatures of Legendre submanifolds Contact δ-invariants δ˜c (n1 , . . . , nk ) and applications . K-contact submanifold satisfying the basic equality . .
17.7
Affine hypersurfaces . . . . . . . . . . . . . . . . . . Centroaffine hypersurfaces . . . . . . . . . . . . . . . Graph hypersurfaces . . . . . . . . . . . . . . . . . . A general optimal inequality for affine hypersurfaces A realization problem for affine hypersurfaces . . . . Applications to affine warped product hypersurfaces 17.6.1 Centroaffine hypersurfaces . . . . . . . . . . 17.6.2 Graph hypersurfaces . . . . . . . . . . . . . Eigenvalues of Tchebychev’s operator KT # . . . . . 17.7.1 Centroaffine hypersurfaces . . . . . . . . . .
305 308 310 318 325 326 328 331 335
. . . . .
. . . . .
Applications to Affine Geometry 17.1 17.2 17.3 17.4 17.5 17.6
279 281 283 286 288 296 298 301 305
Applications to Contact Geometry 16.1 16.2 16.3 16.4 16.5
17.
of δ-invariants to minimal immersions of δ-invariants to spectral geometry . of δ-invariants to homogeneous spaces of δ-invariants to rigidity problems . . to warped products . . . . . . . . . . to Einstein manifolds . . . . . . . . . to conformally flat manifolds . . . . . of δ-invariants to general relativity . .
Applications to K¨ ahler and Para-K¨ahler geometry 15.1 15.2 15.3 15.4 15.5 15.6
16.
Applications Applications Applications Applications Applications Applications Applications Applications
xxix
335 336 338 339 343 345
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346 348 350 351 355 360 360 365 367 368
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Pseudo-Riemannian Geometry, δ-invariants and Applications
17.7.2 Graph hypersurfaces 18.
A submersion δ-invariant . . . . . . . . . . . . . . . An optimal inequality for Riemannian submersions . Some applications . . . . . . . . . . . . . . . . . . . Submersions satisfying the basic equality . . . . . . . A characterization of Cartan hypersurface . . . . . . Links between submersions and affine hypersurfaces
377 . . . . . .
. . . . . .
. . . . . .
Nearly K¨ ahler manifolds and Nearly K¨ahler S 6 (1) 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8
20.
. . . . . . . . . . . . . . . . 374
Applications to Riemannian Submersions 18.1 18.2 18.3 18.4 18.5 18.6
19.
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Real hypersurfaces of nearly K¨ahler manifolds . . . . . . . Nearly K¨ ahler structure on S 6 (1) . . . . . . . . . . . . . . Almost complex submanifolds of nearly K¨ahler manifolds Ejiri’s theorem for Lagrangian submanifolds of S 6 (1) . . . Dillen-Vrancken’s theorem for Lagrangian submanifolds . δ(2) and CR-submanifolds of S 6 (1) . . . . . . . . . . . . . Hopf hypersurfaces of S 6 (1) . . . . . . . . . . . . . . . . . Ideal real hypersurfaces of S 6 (1) . . . . . . . . . . . . . .
δ(2)-ideal Immersions 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12
δ(2)-ideal submanifolds of real space forms . . . . . . . . . δ(2)-ideal tubes in real space forms . . . . . . . . . . . . . δ(2)-ideal isoparametric hypersurfaces in real space forms 2-type δ(2)-ideal hypersurfaces of real space forms . . . . δ(2) and CM C hypersurfaces of real space forms . . . . . δ(2)-ideal conformally flat hypersurfaces . . . . . . . . . . Symmetries on δ(2)-ideal submanifolds . . . . . . . . . . . G2 -structure on S 7 (1) . . . . . . . . . . . . . . . . . . . . δ(2)-ideal associative submanifolds of S 7 (1) . . . . . . . . δ(2)-ideal Lagrangian submanifolds of complex space forms δ(2)-ideal CR-submanifolds of complex space forms . . . . δ(2)-ideal K¨ ahler hypersurfaces in complex space forms .
377 378 381 383 387 389 393 394 397 398 401 403 407 409 413 417 417 419 420 421 422 424 427 429 430 431 435 437
Bibliography
441
GENERAL INDEX
463
AUTHOR INDEX
473
