Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes 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 imbedding 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 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. Differential Geometry of Warped Product Manifolds and Submanifolds Chen Differential Geometry of Warped Product Manifolds and Submanifolds Bang-Yen Chen World Scientific www.worldscientific.com 10419 hc ISBN 978-981-3208-92-6 World Scientific
In memory of Professors S. S. Chern, T. Nagano, T. Otsuki and K. Yano who had the most important influence on my research
Preface Warped products are the most natural and the most fruitful generalization of Cartesian products. More precisely, a warped product is a manifold equipped with a warped product metric of the form: g = i,j g ij (y)dy i dy j +f(y) s,t g st (x)dx s dx t, where the warped geometry decomposes into a product of the y geometry and the x geometry, except that the second part is warped, i.e., it is rescaled by a scalar function of the other coordinates y. If one substitutes the variabley forthe time variabletand xfora3-dimensionalspatialspace, then the first part becomes the effect of time in Einstein s curved space. How it curves space will define one or the other solution to a spacetime model. For that reason different models of spacetime in general relativity are often expressed in terms of warped geometry. Consequently, the notion of warped products plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. The term of warped product was introduced by R. L. Bishop and B. O Neill in [Bishop and O Neill (1964)], who used it to construct a large class of complete manifolds of negative curvature. However, the concept of warped products appeared in the mathematical and physical literature before [Bishop and O Neill (1964)]; for instance, warped products were called semi-reducible spaces in [Kruchkovich (1957)]. Nevertheless, inspired by Bishop and O Neill s article, many important works on warped products from intrinsic point of view were done during the last fifty years. According to the famous Nash embedding theorem published in 1956, every Riemannian manifold can be isometrically embedded in some Euclidean spaces. Nash s theorem shows that every warped product N 1 f N 2 can be embedded as a Riemannian submanifold in some Euclidean spaces xxiii
xxiv Differential Geometry of Warped Product Manifolds and Submanifolds with sufficiently high codimension. Due to this fact, the author asked the following basic question (see, e.g., [Chen (2002a)]). Question: What can we conclude from an isometric immersion of an arbitrary warped product into a Euclidean space or into a space form with arbitrary codimension with arbitrary codimension? The study of warped products from this extrinsic point of view was initiated around the beginning of this century by the author in a series of his articles. Since then the study of warped product submanifolds from extrinsic point of view has become a very active research subject in differential geometry and many nice results on this subject have been obtained by many geometers. The main purpose of this book is thus to provide an extensive and comprehensive survey on the study of warped product manifolds and submanifolds from intrinsic and extrinsic points of view done during the last few decades. It is the author s hope that the reader will find this book both a good introduction to the theories of warped product manifolds and of warped product submanifolds as well as a useful reference for recent and further research of both areas. In concluding the preface, the author would like to thank World Scientific Publishing for the invitation to undertake this project. He also would like to express his appreciation to Professors D. E. Blair, I. Dimitric, O. J. Garay, I. Mihai, M. Petrović-Torgašev, B. Sahin, B. Suceava, J. Van der Veken, and S. W. Wei for reading parts of the manuscript and offering many valuable suggestions. In particular, the author thanks Professor L. Verstraelen for writing an excellent foreword for this book. November 1, 2016 Bang-Yen Chen
Foreword Preface Contents vii 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 Reduction 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 xxv
xxvi Differential Geometry of Warped Product Manifolds and Submanifolds 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
Contents xxvii 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 submanifolds 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
xxviii Differential Geometry of Warped Product Manifolds and Submanifolds 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 CR-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 hypersurfaces in non-flat complex space forms.......................... 300 13.6 Warped product real hypersurfaces............. 314
Contents xxix 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 14.10 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 S 6............... 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 S 6....................... 375 15.8 Non-existence of CR-products in nearly Kähler S 6.... 377 15.9 A special class of warped product submanifolds in nearly Kähler S 6........................... 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 P R-warped products and P-products in para-kähler manifolds........................... 396 16.7 P R-products in non-flat para-kähler space forms..... 398 16.8 Warped product PR-submanifolds............. 400 16.9 P R-warped products satisfying the basic equality..... 406
xxx Differential Geometry of Warped Product Manifolds and Submanifolds 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 hypersurfaces............... 427 18.2 Centroaffine hypersurfaces.................. 429 18.3 Graph hypersurfaces..................... 430 18.4 A realization problem for affine hypersurfaces....... 432 18.5 Warped products as centroaffine hypersurfaces...... 437 18.6 Warped products as graph hypersurfaces.......... 442 18.7 Realization of Robertson-Walker spaces as affine hypersurfaces......................... 443 Bibliography 451 General Index 473 Author Index 481