Differential Geometry - Riemannian Geometry and Isometric Immersions (Book I-B)

Description

This book, Differential Geometry: Riemannian Geometry and Isometric Immersions (Book I-B), is the first in a captivating series of four books presenting a choice of topics, among fundamental and more advanced in differential geometry (DG). Starting with the basics of semi-Riemannian geometry, the book aims to develop the understanding of smooth 1-parameter variations of geodesics of, and correspondingly of, Jacobi fields. A few algebraic aspects required by the treatment of the Riemann–Christoffel four-tensor and sectional curvature are successively presented. Ricci curvature and Einstein manifolds are briefly discussed. The Sasaki metric on the total space of the tangent bundle over a Riemannian manifold is built, and its main properties are investigated. An important integration technique on a Riemannian manifold, related to the geometry of geodesics, is presented for further applications. The other three books of the series are Differential Geometry 1: Manifolds, Bundle and Characteristic Classes (Book I-A) Differential Geometry 3: Foundations of Cauchy-Riemann and Pseudohermitian Geometry (Book I-C) Differential Geometry 4: Advanced Topics in Cauchy–Riemann and Pseudohermitian Geometry (Book I-D) The four books belong to a larger book project (Differential Geometry, Partial Differential Equations, and Mathematical Physics) by the same authors, aiming to demonstrate how certain portions of DG and the theory of partial differential equations apply to general relativity and (quantum) gravity theory. These books supply some of the ad hoc DG machinery yet do not constitute a comprehensive treatise on DG, but rather authors’ choice based on their scientific (mathematical and physical) interests. These are centered around the theory of immersions—isometric, holomorphic, Cauchy–Riemann (CR)—and pseudohermitian geometry, as devised by Sidney Martin Webster for the study of nondegenerate CR structures, themselves a DG manifestation of the tangential CR equations. Preface to Book I-B Contents About the Authors 1 Riemannian Geometry 1.1 Riemannian and Semi-Riemannian Metrics 1.1.1 Historical Note 1.1.2 Bilinear Symmetric Forms 1.1.3 Inner Products 1.1.4 Semi-Riemannian Manifolds 1.1.5 Canonical Measure 1.1.6 Levi–Civita Connection 1.2 Geodesics and Jacobi Fields 1.3 Riemann-Christoffel Tensor Field 1.4 Sectional Curvature 1.5 Ricci Curvature, Einstein Manifolds 1.6 Canonical Structures on the Tangent Bundle 1.7 Geodesics and Integration 1.8 Laplace–Beltrami Operator 1.8.1 Bochner–Lichnerowicz Formula 1.8.2 Laplacian in Radial Coordinates 1.9 Hodge–de Rham Theory 1.9.1 Star Isomorphism 1.9.2 Harmonic Forms 1.9.3 Orthogonality Relations 1.9.4 Hodge–de Rham Decomposition Theorem 1.9.5 The Operators dd, \deltaδ and \DeltaΔ in Terms of Covariant Derivatives 1.9.6 Green's Operator 1.9.7 Killing Vector Fields 1.10 Compact Lie Groups 1.10.1 Grassman Algebra of a Lie Group 1.10.2 Invariant Differential Forms 1.10.3 Compact Semi-simple Lie Groups 1.10.4 Harmonic Forms on Compact Semi-simple Lie Groups 1.11 Curvature and Fundamental Groups 1.12 Kanai Graphs 1.12.1 Quasi-isometries, Pseudo-isometries, and Rough Isometries 1.12.2 Nets in Riemannian Geometry 1.12.3 Volume Growth Rate 1.12.4 Isoperimetric Constants 1.12.5 Liouville's Theorem 1.12.6 Rough Isometries and Parabolicity of Riemannian Manifolds 1.12.7 Rough Isometries and Harmonic Functions on Graphs 1.12.8 Parabolic and Hyperbolic Infinite Networks 1.12.9 Parabolic Index and Rough Isometries 1.12.10 Appendix: Martin Boundary in Potential Theory References 2 Finslerian Geometry 2.1 Finslerian Geometry 2.1.1 Pullback Bundle 2.1.2 The Vertical Lift 2.1.3 Connections in the Pullback Bundle 2.1.4 Nonlinear Connections 2.1.5 Finslerian Connections 2.1.6 Finslerian Metrics 2.1.7 Cartan Connection 2.2 Geodesics in Finslerian Manifolds 2.2.1 Regular Curves 2.2.2 Parametrized Surfaces in upper V left parenthesis upper M right parenthesisV(M) 2.2.3 Existence and Uniqueness of Geodesics in Finslerian Manifolds 2.2.4 Geodesics with Given Initial Data 2.2.5 Applying Banach's Fixed Point Theorem 2.2.6 Convex Neighborhoods 2.3 Finslerian Distance Functions 2.4 Complete Finslerian Manifolds 2.5 First and Second Variations of the Energy Function 2.5.1 The Energy Function 2.5.2 Critical Points of the Energy Function 2.6 A Postface on Finslerian Geometry References 3 Isometric Immersions 3.1 Fundamental Forms of an Immersed Manifold 3.2 Normal Bundle 3.3 Gauss and Weingarten Formulas 3.4 Covariant Calculus in normal upper Omega Superscript 0 Baseline left parenthesis f Superscript negative 1 Baseline upper T left parenthesis ModifyingAbove upper M With quotation dash right parenthesis right parenthesisΩ0 ( f-1 T(overlineM)) 3.5 Gauss–Ricci–Codazzi Equations 3.6 Totally Geodesic and Minimal Submanifolds 3.7 Cartan's Theorem 3.8 Nomizu's Theorem 3.9 Submanifolds of Brieskorn Spheres 3.10 Topology of Brieskorn Spheres 3.11 Germs of Totally Geodesic Submanifolds 3.12 Minimal Submanifolds 3.12.1 Basic Notions and Results, Examples 3.12.2 Minimal Surfaces in double struck upper R Superscript nmathbbRn 3.12.3 Weierstrass Representation of Minimal Surfaces 3.12.4 Gauss Map of a Minimal Surface 3.12.5 Minimal Submanifolds of Spheres 3.13 Harmonic Maps 3.13.1 Harmonic Maps of Spheres 3.13.2 Fundamental Problems in the Theory of Harmonic Maps 3.13.3 Lichnerowicz Theorem 3.13.4 Harmonic Maps of Surfaces 3.13.5 Isoparametric Functions 3.14 Submanifolds of Space Forms 3.15 Chern's Theorem 3.16 Allendoerfer–Erbacher's Theorem 3.17 Totally Umbilical Submanifolds 3.18 Characterizations of the Sphere 3.18.1 Hypersurfaces in Euclidean Spaces 3.18.2 Real Hypersurfaces in Euclidean Complex Space Forms 3.18.3 Compact Submanifolds in Euclidean Spaces 3.19 Submanifolds of Spheres 3.19.1 Hypersurfaces in Spheres 3.19.2 Small Spheres in Spheres 3.19.3 Minimal Submanifolds in Spheres 3.19.4 Biharmonic Immersions in Spheres 3.20 Calabi's Theorem References 4 Riemannian Foliations 4.1 Transverse Riemannian Structures, Transverse Metrics, Bundle-Like Metrics 4.2 Bott Connection, Canonical Adapted Connection 4.3 Second Fundamental Form and Mean Curvature 4.4 d Subscript Script Upper FdmathscrF-Cohomology 4.5 Atiyah Class 4.6 A Spectral Sequence 4.6.1 Filtrations of the De Rham Complex and Rummler's Formula Revisited 4.6.2 Appendix: Spectral Sequences 4.7 Foliations by Level Hypersurfaces 4.8 De Rham-Hodge Theory for Riemannian Foliations 4.8.1 Alaoui–Hector–Sergiescu Foliated De Rham Decomposition Theorem 4.8.2 Exterior Differential Calculus on normal Upper Omega Superscript Bullet Baseline Left Parenthesis Upper M Right ParenthesisΩ(M) Vversus normal upper Omega Subscript upper B Superscript bullet Baseline left parenthesis script upper F right parenthesisΩB (mathscrF) 4.8.3 Sobolev Chain on normal Upper Omega Subscript Upper B Superscript Bullet Baseline Left Parenthesis Script Upper F Right ParenthesisΩB (mathscrF) 4.8.4 Coercivity and Weak Solvability 4.8.5 Regularity 4.8.6 Proof of Alaoui–Hector–Sergiescu Theorem 4.9 Riemannian Foliations and Curvature Groups 4.9.1 The Basic Vaisman Complex 4.9.2 Infinitesimal Conformal Transformations 4.9.3 The Relationship to Basic Cohomology 4.9.4 Basic Curvature Groups of Riemannian Foliations Modeled on Space Forms 4.10 Foliated Harmonic Maps 4.10.1 Lichnerowicz Type Theorem on Transversally Holomorphic Maps 4.10.2 left Parenthesis Script Upper F Comma Script Upper G Right Parenthesis(mathscrF, mathscrG)-Harmonic Maps of Foliated Riemannian Manifolds References 5 Sobolev Inequalities on Submanifolds 5.1 Sobolev Inequalities on Complete Manifolds 5.1.1 Sobolev Inequalities on Euclidean Space 5.1.2 Sobolev Inequalities on Complete Riemannian Manifolds with Ricci Curvature Bounded from Below 5.1.3 Poincaré Inequality 5.1.4 Sobolev Inequalities on Complete Riemannian Manifolds 5.2 Sobolev Inequalities on Manifolds of Negative Curvature 5.3 Sobolev Inequalities with Euclidean Best Constant 5.4 Hoffman–Spruck Theorem 5.5 Brendle's Theorems 5.6 Batista–Mirandola Theorem References Index