Pseudo-Riemannian Geometry, Delta-Invariants and Applications

by Chen, Bang-Yen

• ISBN13: 9789814329637
• ISBN10: 9814329630
• Format: Hardcover
• Copyright: 2011-06-30
• Publisher: World Scientific Pub Co Inc

Summary

The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian 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 ?-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 ?-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 ?-invariants and the main extrinsic invariants. Since then many new results concerning these ?-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.

Table of Contents

Preface	p. vii
Foreword	p. ix
Pseudo-Riemannian Manifolds	p. 1
Symmetric bilinear forms and scalar products	p. 2
Pseudo-Riemannian manifolds	p. 3
Physical interpretations of pseudo-Riemannian manifolds	p. 5
4D spacetimes	p. 5
Kaluza-Klein theory and pseudo-Riemannian manifolds of higher dimension	p. 7
Levi-Civita connection	p. 9
Parallel translation	p. 11
Riemann curvature tensor	p. 15
Sectional, Ricci and scalar curvatures	p. 17
Indefinite real space forms	p. 19
Lie derivative, gradient, Hessian and Laplacian	p. 21
Weyl conformal curvature tensor	p. 24
Basics on Pseudo-Riemannian Submanifolds	p. 25
Isometric immersions	p. 26
Cartan-Janet's and Nash's embedding theorems	p. 27
Gauss' formula and second fundamental form	p. 28
Weingarten's formula and normal connection	p. 30
Shape operator of pseudo-Riemannian submanifolds	p. 33
Fundamental equations of Gauss, Codazzi and Ricci	p. 34
Fundamental theorems of submanifolds	p. 38
A reduction theorem of Erbacher-Magid	p. 39
Two basic formulas for submanifolds in Ems	p. 41
Relationship between squared mean curvature and Ricci curvature	p. 44
Relationship between shape operator and Ricci curvature	p. 47
Cartan's structure equations	p. 52
Special Pseudo-Riemannian Submanifolds	p. 53
Totally geodesic submanifolds	p. 53
Parallel submanifolds of (indefinite) real space forms	p. 55
Totally umbilical submanifolds	p. 57
Totally umbilical submanifolds of Sms(1) and Hms(-1)	p. 60
Pseudo-umbilical submanifolds of Ems	p. 63
Pseudo-umbilical submanifolds of Sms(l) and Hms(-1)	p. 64
Minimal Lorentz surfaces in indefinite real space forms	p. 67
Marginally trapped surfaces and black holes	p. 71
Quasi-minimal surfaces in indefinite space forms	p. 75
Warped Products and Twisted Products	p. 77
Basics of warped products	p. 78
Curvature of warped products	p. 80
Warped product immersions	p. 83
Twisted products	p. 86
Double-twisted products and their characterization	p. 89
Robertson-Walker Spacetimes	p. 91
Cosmology, Robertson-Walker spacetimes and Einstein's field equations	p. 91
Basic properties of Robertson-Walker spacetimes	p. 94
Totally geodesic submanifolds of RW spacetimes	p. 98
Parallel submanifolds of RW spacetimes	p. 99
Totally umbilical submanifolds of RW spacetimes	p. 101
Hypersurfaces of constant curvature in RW spacetimes	p. 105
Realization of RW spacetimes in pseudo-Euclidean spaces	p. 106
Hodge Theory, Elliptic Differential Operators and Jacobi's Elliptic Functions	p. 107
Operators d, * and ¿	p. 108
Hodge-Laplace operator	p. 111
Elliptic differential operator	p. 112
Hodge-de Rham decomposition and its applications	p. 115
The fundamental solution of heat equation	p. 117
Spectra of some important Riemannian manifolds	p. 120
Spectra of flat tori	p. 124
Heat equation, Jacobi's elliptic and theta functions	p. 125
Submanifolds of Finite Type	p. 127
Order and type of submanifolds	p. 128
Minimal polynomial criterion	p. 131
A variational minimal principle	p. 134
Classification of 1-type submanifolds	p. 137
Finite type immersions of compact homogeneous spaces	p. 138
Submanifolds of Ems satisfying ¿H = ¿H	p. 140
Submanifolds of Hm<(-1) satisfying ¿H = ¿H	p. 142
Submanifolds of Sm1(l) satisfying ¿H = ¿H	p. 144
Biharmonic submanifolds	p. 145
Null 2-type submanifolds	p. 148
Spherical 2-type submanifolds	p. 152
2-type hypersurfaces in hyperbolic spaces	p. 156
Total Mean Curvature	p. 161
Total mean curvature of tori in E3	p. 162
Total mean curvature and conformal invariants	p. 164
Total mean curvature for arbitrary submanifolds	p. 167
Total mean curvature and order of submanifolds	p. 171
Conformal property of ¿ 1vol(M)	p. 175
Total mean curvature and ¿ 1,¿ 2	p. 176
Total mean curvature and circumscribed radii	p. 178
Pseudo-Kahler Manifolds	p. 183
Pseudo-Kähler manifolds	p. 184
Pseudo-Kähler submanifolds	p. 187
Purely real submanifolds of pseudo-Kähler manifolds	p. 190
Dependence of fundamental equations for Lorentz surfaces	p. 192
Totally real and Lagrangian submanifolds	p. 196
CR-submanifolds of pseudo-Kähler manifolds	p. 198
Slant submanifolds of pseudo-Kähler manifolds	p. 202
Para-Kähler Manifolds	p. 205
Para-Kähler manifolds	p. 206
Para-Kähler space forms	p. 207
Invariant submanifolds of para-Kähler manifolds	p. 209
Lagrangian submanifolds of para-Kähler manifolds	p. 211
Scalar curvature of Lagrangian submanifolds	p. 214
Ricci curvature of Lagrangian submanifolds	p. 216
Lagrangian H-umbilical submanifolds	p. 218
'P R-submanifolds of para-Kähler manifolds	p. 221
Pseudo-Riemannian Submersions	p. 227
Pseudo-Riemannian submersions	p. 228
O'Neill integrability tensor and O'Neill's equations	p. 229
Submersions with totally geodesic fibers	p. 230
Submersions with minimal fibers	p. 234
A cohomology class for Riemannian submersion	p. 237
Geometry of horizontal immersions	p. 239
Contact Metric Manifolds and Submanifolds	p. 241
Contact pseudo-Riemannian metric manifolds	p. 242
Sasakian manifolds	p. 242
Sasakian space forms with definite metric	p. 244
Sasakian space forms with indefinite metric	p. 245
Legendre submanifolds via canonical fibration	p. 247
Contact slant submanifolds via canonical fibration	p. 249
¿-Invariants, Inequalities and Ideal Immersions	p. 251
Motivation	p. 251
Definition of ¿-invariants	p. 252
¿-invariants and Einstein and conformally flat manifolds	p. 254
Fundamental inequalities involving ¿-invariants	p. 260
Ideal immersions via ¿-invariants	p. 268
Examples of ideal immersions	p. 270
¿-invariants of curvature-like tensor	p. 271
A dimension and decomposition theorem	p. 275
Some Applications of ¿-invariants	p. 279
Applications of ¿-invariants to minimal immersions	p. 279
Applications of ¿-invariants to spectral geometry	p. 281
Applications of ¿-invariants to homogeneous spaces	p. 283
Applications of ¿-invariants to rigidity problems	p. 286
Applications to warped products	p. 288
Applications to Einstein manifolds	p. 296
Applications to conformally flat manifolds	p. 298
Applications of ¿-invariants to general relativity	p. 301
Applications to Kähler and Para-Kähler geometry	p. 305
A vanishing theorem for Lagrangian immersions	p. 305
Obstructions to Lagrangian isometric immersions	p. 308
Improved inequalities for Lagrangian submanifolds	p. 310
Totally real ¿-invariants ¿rk and their applications	p. 318
Examples of strongly minimal Kähler submanifolds	p. 325
Kählerian ¿-invariants ¿c and their applications to Kähler submanifolds	p. 326
Applications of ¿-invariants to real hypersurfaces	p. 328
Applications of ¿-invariants to para-Kähler manifolds	p. 331
Applications to Contact Geometry	p. 335
¿-invariants and submanifolds of Sasakian space forms	p. 335
¿-invariants and Legendre submanifolds	p. 336
Scalar and Ricci curvatures of Legendre submanifolds	p. 338
Contact ¿-invariants ¿c(n1,…,nk) and applications	p. 339
K-contact submanifold satisfying the basic equality	p. 343
Applications to Affine Geometry	p. 345
Affine hypersurfaces	p. 346
Centroaffine hypersurfaces	p. 348
Graph hypersurfaces	p. 350
A general optimal inequality for affine hypersurfaces	p. 351
A realization problem for affine hypersurfaces	p. 355
Applications to affine warped product hypersurfaces	p. 360
Centroaffine hypersurfaces	p. 360
Graph hypersurfaces	p. 365
Eigenvalues of Tchebychev's operator KT#	p. 367
Centroaffine hypersurfaces	p. 368
Graph hypersurfaces	p. 374
Applications to Riemannian Submersions	p. 377
A submersion ¿-invariant	p. 377
An optimal inequality for Riemannian submersions	p. 378
Some applications	p. 381
Submersions satisfying the basic equality	p. 383
A characterization of Cartan hypersurface	p. 387
Links between submersions and affine hypersurfaces	p. 389
Nearly Kähler Manifolds and Nearly Kähler S6(1)	p. 393
Real hypersurfaces of nearly Kähler manifolds	p. 394
Nearly Kähler structure on S6(1)	p. 397
Almost complex submanifolds of nearly Kähler manifolds	p. 398
Ejiri's theorem for Lagrangian submanifolds of S6(1)	p. 401
Dillen-Vrancken's theorem for Lagrangian submanifolds	p. 403
¿(2) and C R-submanifolds of S6(1)	p. 407
Hopf hypersurfaces of S6(1)	p. 409
Ideal real hypersurfaces of S6(1)	p. 413
¿(2)-ideal Immersions	p. 417
¿(2)-ideal submanifolds of real space forms	p. 417
¿(2)-ideal tubes in real space forms	p. 419
¿(2)-ideal isoparametric hypersurfaces in real space forms	p. 420
2-type ¿(2)-ideal hypersurfaces of real space forms	p. 421
¿(2) and C M C hypersurfaces of real space forms	p. 422
¿(2)-ideal conformally flat hypersurfaces	p. 424
Symmetries on ¿(2)-ideal submanifolds	p. 427
G2-structure on S7(1)	p. 429
¿(2)-ideal associative submanifolds of S7(1)	p. 430
¿(2)-ideal Lagrangian submanifolds of complex space forms	p. 431
¿(2)-ideal C R-submanifolds of complex space forms	p. 435
¿(2)-ideal Kähler hypersurfaces in complex space forms	p. 437
Bibliography	p. 439
General Index	p. 463
Author Index	p. 473
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