Differential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.

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1 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 hc ISBN World Scientific

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3 In memory of Professors S. S. Chern, T. Nagano, T. Otsuki and K. Yano who had the most important influence on my research

4 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

5 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

6 Foreword Preface Contents vii xxiii 1. Riemannian and Pseudo-Riemannian Manifolds Symmetric bilinear forms and scalar products Riemannian and pseudo-riemannian manifolds Levi-Civita connection Parallel transport Riemann curvature tensor Sectional, Ricci and scalar curvatures Indefinite real space forms Gradient, Hessian and Laplacian Lie derivative and Killing vector fields Concircular and concurrent vector fields Submanifolds Embedding theorems Formulas of Gauss and Weingarten Equations of Gauss, Codazzi and Ricci Existence and uniqueness theorems of submanifolds Reduction theorems Totally geodesic submanifolds Totally umbilical submanifolds Pseudo-umbilical submanifolds Cartan s structure equations xxv

7 xxvi Differential Geometry of Warped Product Manifolds and Submanifolds 3. Warped Product Manifolds Warped products Connection of warped products Curvature of warped products Einstein warped product manifolds Conformally flat warped product manifolds Multiply warped product manifolds Warped product immersions More results for warped product immersions Twisted products Characterizations of twisted products Convolution manifolds Robertson-Walker Spacetimes and Schwarzschild Solution Basic properties of Robertson-Walker spacetimes Totally geodesic submanifolds of Robertson-Walker spacetimes Parallel submanifolds of Robertson-Walker spacetimes Totally umbilical submanifolds of Robertson-Walker spacetimes Realizations of Robertson-Walker spacetimes Generalized Robertson-Walker spacetimes Schwarzschild s solution and black holes Contact Metric Manifolds and Submersions Contact metric manifolds Sasakian manifolds Submersions O Neill integrability tensor and fundamental equations Submersions with totally geodesic fibers Sasakian space forms Geometry of horizontal immersions Legendre submanifolds via canonical fibration Kähler and Pseudo-Kähler Manifolds Pseudo-Kähler manifolds Concircular vector fields on pseudo-kähler manifolds Pseudo-Kähler submanifolds

8 Contents xxvii 6.4 Segre and Veronese embeddings Purely real submanifolds of pseudo-kähler manifolds Totally real and Lagrangian submanifolds Totally umbilical and H-umbilical submanifolds Warped products, H-umbilical submanifolds and complex extensors Classification of H-umbilical submanifolds Slant Submanifolds Examples of slant submanifolds Basic properties and their applications Existence and uniqueness theorems A non-existence theorem for compact slant submanifolds A non-minimality theorem for slant submanifolds Topology and cohomology of slant submanifolds Pointwise slant submanifolds Contact slant submanifolds via canonical fibration Generic Submanifolds of Kähler Manifolds Generic submanifolds Integrability Parallelism of P and F Totally umbilical submanifolds Generic products and Segre embedding Generic products in complex projective spaces An application to complex geometry CR-submanifolds of Kähler Manifolds CR-submanifolds as CR-manifolds Integrability and minimality Cohomology of CR-submanifolds Totally geodesic and totally umbilical CR-submanifolds Mixed foliate CR-submanifolds Warped Products in Riemannian and Kähler Manifolds An algebraic lemma Warped products in real space forms

9 xxviii Differential Geometry of Warped Product Manifolds and Submanifolds 10.3 Some applications of Theorems 10.1 and Rotation hypersurfaces in real space forms Another optimal inequality for warped products Warped products in Kähler manifolds Warped product submanifolds in generalized complex space forms Warped Product Submanifolds of Kähler Manifolds Warped product CR-submanifolds CR-warped products and their characterization Examples of CR-warped products A general inequality for CR-warped products Twisted product CR-submanifolds Warped product submanifolds with a holomorphic factor Warped product hemi-slant submanifolds Warped product semi-slant submanifolds Warped product pointwise semi-slant submanifolds Warped product pointwise bi-slant submanifolds Warped products in locally conformal Kähler manifolds CR-warped Products in Complex Space Forms CR-warped products A PDE system associated with the basic equality CR-warped products in C m satisfying basic equality CR-warped products in CP m and CH m CR-warped products with compact holomorphic factor More on CR-warped Products in Complex Space Forms Another optimal inequality for CR-warped products CR-warped products in C m satisfying the equality CR-warped products in CP m satisfying the equality CR-warped products in CH m satisfying the equality Irreducibility of real hypersurfaces in non-flat complex space forms Warped product real hypersurfaces

10 Contents xxix 14. δ-invariants, Submersions and Warped Products δ-invariants An inequality for submanifolds in real space forms Inequalities for submanifolds in complex space forms Improved inequalities for Lagrangian submanifolds CR-warped products and δ-invariants Anti-holomorphic submanifolds with p Anti-holomorphic submanifolds satisfying the equality An optimal inequality for real hypersurfaces Another optimal inequality involving a δ-invariant Examples of δ(2)-ideal warped product submanifolds Warped Products in Nearly Kähler Manifolds Nearly Kähler manifolds Nearly Kähler structure on S Complex submanifolds of nearly Kähler manifolds Lagrangian submanifolds of nearly Kähler manifolds CR-submanifolds in nearly Kähler manifolds Warped products in nearly Kähler manifolds Examples of warped product CR-submanifolds in nearly Kähler S Non-existence of CR-products in nearly Kähler S A special class of warped product submanifolds in nearly Kähler S Warped Products in Para-Kähler Manifolds Para-Kähler manifolds Non-flat para-kähler space forms Invariant submanifolds of para-kähler manifolds Lagrangian submanifolds of para-kähler manifolds PR-submanifolds in para-kähler manifolds P R-warped products and P-products in para-kähler manifolds P R-products in non-flat para-kähler space forms Warped product PR-submanifolds P R-warped products satisfying the basic equality

11 xxx Differential Geometry of Warped Product Manifolds and Submanifolds 17. Warped Products in Sasakian Manifolds Sasakian manifolds and submanifolds Warped products in Sasakian manifolds Contact CR-submanifolds CR-warped products with smallest codimension Another inequality for contact CR-warped products in Sasakian manifolds Pointwise bi-slant and hemi-slant warped products in Sasakian manifolds Warped Products in Affine Spaces Affine spaces and hypersurfaces Centroaffine hypersurfaces Graph hypersurfaces A realization problem for affine hypersurfaces Warped products as centroaffine hypersurfaces Warped products as graph hypersurfaces Realization of Robertson-Walker spaces as affine hypersurfaces Bibliography 451 General Index 473 Author Index 481