DIFFERENTIAL GEOMETRY WARPED PRODUCT MANIFOLD & SUBMANIFOLD

by Bang-yen Chen

Overview

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson-Walker models, are warped product manifolds.The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson-Walker's and Schwarzschild's.The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.

Product Details

ISBN-13:	9789813208940
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/29/2017
Sold by:	Barnes & Noble
Format:	eBook
Pages:	516
File size:	25 MB
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Table of Contents

Foreword vii

Preface xxiii

1 Riemannian and Pseudo-Riemannian Manifolds 1

1.1 Symmetric bilinear forms and scalar products 1

1.2 Riemannian and pseudo-Riemannian manifolds 3

1.3 Levi-Civita connection 4

1.4 Parallel transport 7

1.5 Riemann curvature tensor 10

1.6 Sectional, Ricci and scalar curvatures 12

1.7 Indefinite real space forms 15

1.8 Gradient, Hessian and Laplacian 16

1.9 Lie derivative and Killing vector fields 17

1.10 Concircular and concurrent vector fields 19

2 Submanifolds 23

2.1 Embedding theorems 24

2.2 Formulas of Gauss and Weingarten 26

2.3 Equations of Gauss, Codazzi and Ricci 30

2.4 Existence and uniqueness theorems of submanifolds 34

2.5 Redaction theorems 35

2.6 Totally geodesic submanifolds 37

2.7 Totally umbilical submanifolds 38

2.8 Pseudo-umbilical submanifolds 41

2.9 Cartan's structure equations 46

3 Warped Product Manifolds 47

3.1 Warped products 47

3.2 Connection of warped products 49

3.3 Curvature of warped products 50

3.4 Einstein warped product manifolds 52

3.5 Conformally flat warped product manifolds 58

3.6 Multiply warped product manifolds 59

3.7 Warped product immersions 62

3.8 More results for warped product immersions 65

3.9 Twisted products 71

3.10 Characterizations of twisted products 76

3.11 Convolution manifolds 78

4 Robertson-Walker Spacetimes and Schwarzschild Solution 81

4.1 Basic properties of Robertson-Walker spacetimes 82

4.2 Totally geodesic submanifolds of Robertson-Walker spacetimes 86

4.3 Parallel submanifolds of Robertson-Walker spacetimes 87

4.4 Totally umbilical submanifolds of Robertson-Walker spacetimes 89

4.5 Realizations of Robertson-Walker spacetimes 93

4.6 Generalized Robertson-Walker spacetimes 94

4.7 Schwarzschild's solution and black holes 96

5 Contact Metric Manifolds and Submersions 99

5.1 Contact metric manifolds 100

5.2 Sasakian manifolds 100

5.3 Submersions 102

5.4 O'Neill integrability tensor and fundamental equations 103

5.5 Submersions with totally geodesic fibers 105

5.6 Sasakian space forms 107

5.7 Geometry of horizontal immersions 111

5.8 Legendre submanifolds via canonical fibration 112

6 Kähler and Pseudo-Kähler Manifolds 115

6.1 Pseudo-Kähler manifolds 115

6.2 Concircular vector fields on pseudo-Kähler manifolds 119

6.3 Pseudo-Kähler submanifolds 121

6.4 Segre and Veronese embeddings 124

6.5 Purely real submanifolds of pseudo-Kähler manifolds 125

6.6 Totally real and Lagrangian submanifolds 127

6.7 Totally umbilical and H-umbilical submanifolds 129

6.8 Warped products, H-umbilical submanifolds and complex extensors 131

6.9 Classification of H-umbilical submanifolds 134

7 Slant Submanifolds 141

7.1 Examples of slant submanifolds 141

7.2 Basic properties and their applications 144

7.3 Existence and uniqueness theorems 151

7.4 A non-existence theorem for compact slant submanifolds 158

7.5 A non-minimality theorem for slant submanifolds 162

7.6 Topology and cohomology of slant submanifolds 165

7.7 Pointwise slant submanifolds 171

7.8 Contact slant submariifolds via canonical fibration 177

8 Generic Submanifolds of Kähler Manifolds 179

8.1 Generic submanifolds 179

8.2 Integrability 181

8.3 Parallelism of P and F 182

8.4 Totally umbilical submanifolds 187

8.5 Generic products and Segre embedding 190

8.6 Generic products in complex projective spaces 191

8.7 An application to complex geometry 193

9 CR-submanifolds of Kähler Manifolds 195

9.1 CR-submanifolds as CR-manifolds 195

9.2 Integrability and minimality 197

9.3 Cohomology of CR-submanifolds 200

9.4 Totally geodesic and totally umbilical CR-submanifolds 202

9.5 Mixed foliate CR-submanifolds 205

10 Warped Products in Riemannian and Kähler Manifolds 207

10.1 An algebraic lemma 207

10.2 Warped products in real space forms 209

10.3 Some applications of Theorems 10.1 and 10.2 213

10.4 Rotation hypersurfaces in real space forms 215

10.5 Another optimal inequality for warped products 217

10.6 Warped products in Kähler manifolds 222

10.7 Warped product submanifolds in generalized complex space forms 227

11 Warped Product Submanifolds of Kähler Manifolds 229

11.1 Warped product CR-submanifolds 229

11.2 OR-warped products and their characterization 231

11.3 Examples of CR-warped products 233

11.4 A general inequality for CR-warped products 235

11.5 Twisted product CR-submanifolds 238

11.6 Warped product submanifolds with a holomorphic factor 242

11.7 Warped product hemi-slant submanifolds 244

11.8 Warped product semi slant submanifolds 248

11.9 Warped product pointwise semi-slant submanifolds 251

11.10 Warped product pointwise bi-slant submanifolds 252

11.11 Warped products in locally conformal Kähler manifolds 254

12 CR-warped Products in Complex Space Forms 257

12.1 CR-warped products 257

12.2 A PDE system associated with the basic equality 259

12.3 CR-warped products in C^m satisfying basic equality 262

12.4 CR-warped products in CP^m and CH^m 270

12.5 CR-warped products with compact holomorphic factor 276

13 More on CR-warped Products in Complex Space Forms 283

13.1 Another optimal inequality for CR-warped products 283

13.2 CR-warped products in C^m satisfying the equality 286

13.3 CR-warped products in CP^m satisfying the equality 296

13.4 CR-warped products in CH^m satisfying the equality 299

13.5 Irreducibility of real hypersurfaees in non-flat complex space forms 300

13.6 Warped product real hypersurfaces 314

14 δ-invariants, Submersions and Warped Products 325

14.1 δ-invariants 326

14.2 An inequality for submanifolds in real space forms 327

14.3 Inequalities for submanifolds in complex space forms 332

14.4 Improved inequalities for Lagrangian submanifolds 336

14.5 CR-warped products and δ-invariants 338

14.6 Anti-holomorphic submanifolds with p ≥ 2 341

14.7 Anti-holomorphic submanifolds satisfying the equality 344

14.8 An optimal inequality for real hypersurfaces 346

14.9 Another optimal inequality involving a δ-invariant 349

11.1 Examples of δ(2)-ideal warped product submanifolds 355

15 Warped Products in Nearly Kähler Manifolds 359

15.1 Nearly Kähler manifolds 359

15.2 Nearly Kähler structure on S⁶ 361

15.3 Complex submanifolds of nearly Kähler manifolds 363

15.4 Lagrangian submanifolds of nearly Kähler manifolds 366

15.5 CR-submanifolds in nearly Kähler manifolds 370

15.6 Warped products in nearly Kähler manifolds 372

15.7 Examples of warped product CR-submanifolds in nearly Kähler S⁶ 375

15.8 Non-existence of CR-products in nearly Kähler S⁶ 377

15.9 A special class of warped product submanifolds in nearly Kähler S⁶ 380

16 Warped Products in Para-Kähler Manifolds 383

16.1 Para-Kähler manifolds 383

16.2 Non-flat para-Kähler space forms 385

16.3 Invariant submanifolds of para-Kähler manifolds 387

16.4 Lagrangian submanifolds of para-Kähler manifolds 389

16.5 PR-submanifolds in para-Kähler manifolds 392

16.6 PR-warped products and P-products in para-Kähler manifolds 396

16.7 PR-products in non-flat para-Kähler space forms 398

16.8 Warped product PR-submanifolds 400

16.9 PR-warped products satisfying the basic equality 406

17 Warped Products in Sasakian Manifolds 409

17.1 Sasakian manifolds and submanifolds 409

17.2 Warped products in Sasakian manifolds 412

17.3 Contact CR-submanifolds 414

17.4 CR-warped products with smallest codimension 418

17.5 Another inequality for contact CR-warped products in Sasakian manifolds 420

17.6 Pointwise bi-slant and hemi-slant warped products in Sasakian manifolds 424

18 Warped Products in Affine Spaces 427

18.1 Affine spaces and hypersurfaccs 427

18.1 Centroaffine hypersurfaces 429

18.1 Graph hypersurfaees 430

18.2 A realization problem for affine hypersurfaces 432

18.3 Warped products as centroaffine hyper surf aces 437

18.4 Warped products as graph hyper surfaces 442

18.5 Realization of Robertson-Walker spaces as affine hypersurfaces 443

Bibliography 451

General Index 473

Author Index 481