This book is an introduction to the concepts, major results and techniques in quintessential Riemannian Geometry. All the concepts are geometrically motivated. Concepts are explained with a set of important examples. The discerning reader will find some unusual examples that are usually hard to find in a book of this nature. Variety of techniques are used for proving results and for carrying out computations. This will enable the reader to consolidate his/her understanding of various concepts and results. The importance of Jacobi fields, variational and analytical methods are emphasized throughout. The detailed computational steps given in this book may be of use to both theoretical physicists and mathematical analysts. The book attends to the difficulties of a beginner and at the same time it provides quite a few advanced topics for experts.
Author(s): S. Kumaresan
Publisher: Techno World
Year: 2020
Language: English
Pages: 314
City: Kolkata
Tags: Riemannian Geometry
Cover
Preface
Suggestions for a Basic Course
Contents
Foundations
1.1 Differential Calculus-A Review
1.1.1 Inverse Mapping Theorem
1.1.2 Basic Theorem in ODE
1.2 Differential Manifolds
1.3 Smooth Maps & Diffeomorphisms
1.4 Tangent Spaces to a Manifold
1.5 Derivatives of Smooth Maps
1.6 Immersions and Submersions
1.7 Submanifolds
1.8 Tensor Fields
1.8.1 Vector Fields
1.8.2 Left-Invariant Vector Fields on Lie Groups
1.8.3 Tensor Fields
1.8.4 Lie Derivation
1.9 Orientable Manifolds
Basic Riemannian Geometry
2.1 Covariant Differentiation
2.1.1 Connection on Rn and Hypersurfaces in Rn+1
2.1.2 Connections on Manifolds
2.2 Riemannian Metrics
2.2.1 Riemannian Metrics
2.2.2 Local Isometry and Isometry
2.2.3 Volume Elements and Integration
2.2.4 Inner Metric
2.2.5 Musical Isomorphisms
2.3 The Levi-Civita Connection
2.4 Gauss Theory of Surfaces in R3
2.4.1 Gaussian Curvature
2.4.2 Gauss Theorema Egregium
2.5 Curvature and Parallel Transport
2.5.1 Curvature Operator
2.5.2 Parallel Transport
2.5.3 Holonomy
2.6 The Curvature Tensor
2.6.1 The Riemannian Curvature
2.6.2 Algebraic Theory of Curvature Operators
2.6.3 Definitions of Various Curvatures
2.7 Geodesics
2.7.1 Vector fields along Maps
2.7.2 Two Parameter Maps
2.7.3 Geodesics: Definition and Existence
2.7.4 Examples of Geodesics
2.7.5 Geodesics by Calculus of Variations
2.8 Exponential Map
2.8.1 Exponential Map & Normal Coordinates
2.8.2 Gauss Lemma
2.8.3 Convex Sets
2.9 Riemannian Immersions
2.9.1 Riemannian Submanifolds
2.9.2 Minimal submanifolds and convex hypersurfaces
2.9.3 Totally Geodesic Submanifolds
2.10 Riemannian Submersions
2.11 Hopf-Rinow Theorem
2.12 Cartan-Hadamard Theorem
Variations of Geodesics
3.1 Formulas of Arc-length Variation
3.2 Jacobi Fields
3.2.1 Definition, Examples and Properties
3.2.2 Some Typical and Important Uses of Jacobi Fields
3.2.3 An Easy Comparison Theorem for Jacobi Fields
3.3 Conjugate Points
3.4 Focal Points
3.5 Index Form
3.6 The Morse Index Theorem
3.7 Cut Locus
Techniques
4.1 Rauch-type Comparison Theorems
4.1.1 Rauch Comparison Theorem
4.1.2 Berger Comparison Theorem
4.2 Synge's Technique and Applications
4.2.1 Synge's Theorem
4.2.2 Klingenberg's Injectivity Radius Estimate
4.3 Toponogov's Theorem and Applications
4.3.1 Toponogov's Theorem
4.3.2 Application 1: Maximal Diameter Theorem
4.3.3 Application 2: A Generalized Sphere Theorem
4.4 Volume Comparison Theorem & Applications
4.4.1 Bishop-Gromov Theorem
4.4.2 Application 1: Cheng's Sphere Theorem
4.4.3 Application 2: Growth of Fundamental Group
4.5 Distance functions
4.6 Riccati equations
4.7 Hessian Comparison Theorem
4.8 Paul Levy's Isoperimetric Inequality
4.9 Preismann's Theorem
4.10 The Bochner Technique
Examples
A.l Euclidean Spaces
A.2 Spheres
A.2.1 Cosine Law
A.2.2 Second Fundamental Form
A.2.3 Geodesics
A.2.4 Jacobi Fields
A.3 Hyperbolic Spaces
A.3.1 Hyperbolic Plane
A.3.2 Distance on H
A.3.3 Area of a Hyperbolic Triangle
A.3.4 Gauss Theorem on Excess of Geodesic Triangles
A.3.5 Hyperbolic Spaces
A.3.6 Cosine Law of Hyperbolic Spaces
A.3.7 Other Models
A.3.8 Reflections and Isometries
A.3.9 Curvature of Hyperbolic Spaces
A.3.10 Jacobi Fields on hyperbolic Spaces
A.4 Locally Symmetric Spaces
A.5 Lie Groups
A.6 Surfaces of Revolution
A.6.1 Description and the Metric
A.6.2 Connection and the Christoffel Symbols
A.6.3 Gaussian Curvature
A.6.4 Equations of Geodesics
A.7 Complex Projective Spaces
A.7.1 Geodesics on Pn
A.7.2 Jacobi Fields on Pn and Curvature of Pn
A.7.3 Conjugate and Cut Loci
A.8 Berger's Spheres
A.9 Tangent Bundle
A.9.1 Sasaki Metric
A.9.2 The Geodesic Flow
A.10 The Cotangent Bundle
Connections on Bundles
B.l Cartan structural equations
B.2 Vector Bundles
B.3 Connections in vector bundles
B.4 Forms with Values in a Vector Space
B.5 Forms with Values in a Vector Bundle
B.6 Principal Bundles
B.7 Connections on Principal Bundles
B.8 Curvature Form
References
Index
