Pseudo-Riemannian geometry, Оґ-invariants and applications / Pseudo-Riemannian geometry, [delta]-invariants and applications

This volume constitutes the proceedings of the Hong Kong International Workshop on Statistics in Finance, held at the University of Hong Kong in July 1999. Topics covered include heavy-tailed and nonlinear continuous-time ARMA models for financial time series, and forecast evaluation Pseudo-Riemannian Manifolds; Pseudo-Riemannian Submanifolds; Warped Products and Twisted Products; Robertson-Walker Spacetimes; Hodge Theory and Elliptic Differential Operators; Submanifolds of Finite Type; Total Mean Curvature; Оґ-Invariants, Inequalities; Submanifolds of Kaehler Manifolds; Submanifolds of Para-Kaehler Manifolds; Pseudo-Riemannian Submersions; Affine Hypersurface; Contact Geometry; Applications of Оґ-Invariants; Applications to Relativity 16. Applications to contact geometry. 16.1. [symbol]-invariants and submanifolds of Sasakian space forms. 16.2. [symbol]-invariants and Legendre submanifolds. 16.3. Scalar and Ricci curvatures of Legendre submanifolds. 16.4. Contact [symbol]-invariants [symbol](n[symbol] ..., n[symbol]) and applications. 16.5. K-contact submanifold satisfying the basic equality -- 17. Applications to affine geometry. 17.1. Affine hypersurfaces. 17.2. Centroaffine hypersurfaces. 17.3. Graph hypersurfaces. 17.4. A general optimal inequality for affine hypersurfaces. 17.5. A realization problem for affine hypersurfaces. 17.6. Applications to affine warped product hypersurfaces. 17.7. Eigenvalues of Tchebychev's operator K[symbol] -- 18. Applications to Riemannian submersions. 18.1. A submersion [symbol]-invariant. 18.2. An optimal inequality for Riemannian submersions. 18.3. Some applications. 18.4. Submersions satisfying the basic equality. 18.5. A characterization of Cartan hypersurface. 18.6. Links between submersions and affine hypersurfaces -- 19. Nearly Kahler manifolds and nearly Kahler S[symbol](1). 19.1. Real hypersurfaces of nearly Kahler manifolds. 19.2. Nearly Kahler structure on S[symbol](1). 19.3. Almost complex submanifolds of nearly Kahler manifolds. 19.4. Ejiri's theorem for Lagrangian submanifolds of S[symbol](1). 19.5. Dillen-Vrancken's theorem for Lagrangian submanifolds. 19.6. [symbol](2) and CR-submanifolds of S[symbol](1). 19.7. Hopf hypersurfaces of S[symbol](1). 19.8. Ideal real hypersurfaces of S[symbol](1) -- 20. [symbol](2)-ideal immersions. 20.1. [symbol](2)-ideal submanifolds of real space forms. 20.2. [symbol](2)-ideal tubes in real space forms. 20.3. [symbol](2)-ideal isoparametric hypersurfaces in real space forms. 20.4. 2-type [symbol](2)-ideal hypersurfaces of real space forms. 20.5. [symbol](2) and CMC hypersurfaces of real space forms. 20.6. [symbol](2)-ideal conformally flat hypersurfaces. 20.7. Symmetries on [symbol](2)-ideal submanifolds. 20.8. G[symbol]-structure on S[symbol](1). 20.9. [symbol](2)-ideal associative submanifolds of S[symbol](1). 20.10. [symbol](2)-ideal Lagrangian submanifolds of complex space forms. 20.11. [symbol](2)-ideal CR-submanifolds of complex space forms. 20.12. [symbol](2)-ideal Kahler hypersurfaces in complex space forms 1. Pseudo-Riemannian manifolds. 1.1. Symmetric bilinear forms and scalar products. 1.2. Pseudo-Riemannian manifolds. 1.3. Physical interpretations of pseudo-Riemannian manifolds. 1.4. Levi-Civita connection. 1.5. Parallel translation. 1.6. Riemann curvature tensor. 1.7. Sectional, Ricci and scalar curvatures. 1.8. Indefinite real space forms. 1.9. Lie derivative, gradient, Hessian and Laplacian. 1.10. Weyl conformal curvature tensor -- 2. Basics on pseudo-Riemannian submanifolds. 2.1. Isometric immersions. 2.2. Cartan-Janet's and Nash's embedding theorems. 2.3. Gauss' formula and second fundamental form. 2.4. Weingarten's formula and normal connection. 2.5. Shape operator of pseudo-Riemannian submanifolds. 2.6. Fundamental equations of Gauss, Codazzi and Ricci. 2.7. Fundamental theorems of submanifolds. 2.8. A reduction theorem of Erbacher-Magid. 2.9. Two basic formulas for submanifolds in E[symbol]. 2.10. Relationship between squared mean curvature and Ricci curvature. 2.11. Relationship between shape operator and Ricci curvature. 2.12. Cartan's structure equations -- 3. Special pseudo-Riemannian submanifolds. 3.1. Totally geodesic submanifolds. 3.2. Parallel submanifolds of (indefinite) real space forms. 3.3. Totally umbilical submanifolds. 3.4. Totally umbilical submanifolds of S[symbol] (1) and H[symbol] ( -1). 3.5. Pseudo-umbilical submanifolds of E[symbol]. 3.6. Pseudo-umbilical submanifolds of S[symbol] (1) and H[symbol] ( -1). 3.7. Minimal Lorentz surfaces in indefinite real space forms. 3.8. Marginally trapped surfaces and black holes. 3.9. Quasi-minimal surfaces in indefinite space forms -- 4. Warped products and twisted products. 4.1. Basics of warped products. 4.2. Curvature of warped products. 4.3. Warped product immersions. 4.4. Twisted products. 4.5. Double-twisted products and their characterization -- 5. Robertson-Walker spacetimes. 5.1. Cosmology, Robertson-Walker spacetimes and Einstein's field equations. 5.2. Basic properties of Robertson-Walker spacetimes. 5.3. Totally geodesic submanifolds of RW spacetimes. 5.4. Parallel submanifolds of RW spacetimes. 5.5. Totally umbilical submanifolds of RW spacetimes. 5.6. Hypersurfaces of constant curvature in RW spacetimes. 5.7. Realization of RW spacetimes in pseudo-Euclidean spaces 11. Pseudo-Riemannian submersions. 11. Pseudo-Riemannian submersions. 11.2. O'Neill integrability tensor and O'Neill's equations. 11.3. Submersions with totally geodesic fibers. 11.4. Submersions with minimal fibers. 11.5. A cohomology class for Riemannian submersion. 11.6. Geometry of horizontal immersions -- 12. Contact metric manifolds and submanifolds. 12.1. Contact pseudo-Riemannian metric manifolds. 12.2. Sasakian manifolds. 12.3. Sasakian space forms with definite metric. 12.4. Sasakian space forms with indefinite metric. 12.5. Legendre submanifolds via canonical fibration. 12.6. Contact slant submanifolds via canonical fibration -- 13. [symbol]-invariants, inequalities and ideal immersions. 13.1. Motivation. 13.2. Definition of [symbol]-invariants. 13.3. [symbol]-invariants and Einstein and conformally flat manifolds. 13.4. Fundamental inequalities involving [symbol]-invariants. 13.5. Ideal immersions via [symbol]-invariants. 13.6. Examples of ideal immersions. 13.7. [symbol]-invariants of curvature-like tensor. 13.8. A dimension and decomposition theorem -- 14. Some applications of [symbol]-invariants. 14.1. Applications of [symbol]-invariants to minimal immersions. 14.2. Applications of [symbol]-invariants to spectral geometry. 14.3. Applications of [symbol]-invariants to homogeneous spaces. 14.4. Applications of [symbol]-invariants to rigidity problems. 14.5. Applications to warped products. 14.6. Applications to Einstein manifolds. 14.7. Applications to conformally flat manifolds. 14.8. Applications of [symbol]-invariants to general relativity -- 15. Applications to Kahler and Para-Kahler geometry. 15.1. A vanishing theorem for Lagrangian immersions. 15.2. Obstructions to Lagrangian isometric immersions. 15.3. Improved inequalities for Lagrangian submanifolds. 15.4. Totally real [symbol]-invariants [symbol] and their applications. 15.5. Examples of strongly minimal Kahler submanifolds. 15.6. Kahlerian [symbol]-invariants [symbol] and their applications to Kahler submanifolds. 15.7. Applications of [symbol]-invariants to real hypersurfaces. 15.8. Applications of [symbol]-invariants to para-Kahler manifolds 6. Hodge theory, elliptic differential operators and Jacobi's elliptic functions. 6.1. Operators d, * and [symbol]. 6.2. Hodge-Laplace operator. 6.3. Elliptic differential operator. 6.4. Hodge-de Rham decomposition and its applications. 6.5. The fundamental solution of heat equation. 6.6. Spectra of some important Riemannian manifolds. 6.7. Spectra of flat tori. 6.8. Heat equation, Jacobi's elliptic and theta functions -- 7. Submanifolds of finite type. 7.1. Order and type of submanifolds. 7.2. Minimal polynomial criterion. 7.3. A variational minimal principle. 7.4. Classification of 1-type submanifolds. 7.5. Finite type immersions of compact homogeneous spaces. 7.6. Submanifolds of E[symbol] satisfying [symbol]H = [symbol]H. 7.7. Submanifolds of H[symbol]( -1) satisfying [symbol]H = [symbol]H. 7.8. Submanifolds of S[symbol](1) satisfying [symbol]H = [symbol]H. 7.9. Biharmonic submanifolds. 7.10. Null 2-type submanifolds. 7.11. Spherical 2-type submanifolds. 7.12. 2-type hypersurfaces in hyperbolic spaces -- 8. Total mean curvature. 8.1. Total mean curvature of tori in E[symbol]. 8.2. Total mean curvature and conformal invariants. 8.3. Total mean curvature for arbitrary submanifolds. 8.4. Total mean curvature and order of submanifolds. 8.5. Conformal property of [symbol]vol(M). 8.6. Total mean curvature and [symbol], [symbol]. 8.7. Total mean curvature and circumscribed radii -- 9. Pseudo-Kahler manifolds. 9.1. Pseudo-Kahler manifolds. 9.2. Pseudo-Kahler submanifolds. 9.3. Purely real submanifolds of pseudo-Kahler manifolds. 9.4. Dependence of fundamental equations for Lorentz surfaces. 9.5. Totally real and Lagrangian submanifolds. 9.6. CR-submanifolds of pseudo-Kahler manifolds. 9.7. Slant submanifolds of pseudo-Kahler manifolds -- 10. Para-Kahler manifolds. 10.1. Para-Kahler manifolds. 10.2. Para-Kahler space forms. 10.3. Invariant submanifolds of para-Kahler manifolds. 10.4. Lagrangian submanifolds of para-Kahler manifolds. 10.5. Scalar curvature of Lagrangian submanifolds. 10.6. Ricci curvature of Lagrangian submanifolds. 10.7. Lagrangian H-umbilical submanifolds. 10.8. PR-submanifolds of para-Kahler manifolds 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 [symbol]-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 [symbol]-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 [symbol]-invariants and the main extrinsic invariants. Since then many new results concerning these [symbol]-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 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 d-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 d-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 d-invariants and the main extrinsic invariants. Since then many new results concerning these d-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. 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 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. Front matter 1 Pseudo-Riemannian Manifolds 33 Basics on Pseudo-Riemannian Submanifolds 57 Special Pseudo-Riemannian Submanifolds 85 Warped Products and Twisted Products 109 Robertson–Walker Spacetimes 123 Hodge Theory, Elliptic Differential Operators and Jacobi's Elliptic Functions 139 Submanifolds of Finite Type 159 Total Mean Curvature 193 Pseudo-Kähler Manifolds 215 Para-Kähler Manifolds 237 Pseudo-Riemannian Submersions 259 Contact Metric Manifolds and Submanifolds 273 δ-invariants, Inequalities and Ideal Immersions 283 Some Applications of δ-invariants 311 Applications to Kähler and Para-Kähler Geometry 337 Applications to Contact Geometry 367 Applications to Affine Geometry 377 Applications to Riemannian Submersions 409 Nearly Kähler Manifolds and Nearly Kähler S 6 (1) 425 δ(2)-ideal Immersions 449 Back matter 471 Provides an 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. In this book, a number of results on pseudo-Riemannian submanifolds are also included. Bang-yen Chen. Includes Bibliographical References (p. 439-462) And Indexes.