Notes on Differential Geometry (Mathematics Studies)

✍ Scribed by Noel J. Hicks

• Publisher: D. Van Nostrand Company
• Year: 1965
• Tongue: English
• Leaves: 242
• Edition: TeXromancers
• Category: Library

✦ Table of Contents

Preface to New Typesetting
Preface
Manifolds
Manifolds
Smooth Functions
Vectors and Vector Fields
The Jacobian of a Map
Curves and Integral Curves
Submanifolds
Hypersurfaces of Rn
The Standard Connexion on Rn
The Sphere Map and the Weingarten Map
The Gauss Equation
The Gauss Curvature and Codazzi-Mainardi Equations
Examples
Some Applications
Surfaces in R3
Smoothness and the Neighborhood of a non-Umbilic Point
Surfaces of Constant Curvature
Parallel Surfaces (Normal Maps)
Examples (Surfaces of Revolution)
Lines of Curvature
Tensors and Forms
Tensors and Forms
Connexions
Invariant Viewpoint
Cartan Viewpoint
Coordinate Viewpoint
Difference Tensor of Two Connexions
Bundle Viewpoint
Riemannian Manifolds and Submanifolds
Length and Distance
Riemannian Connexion and Curvature
Curves in Riemannian Manifolds
Submanifolds
Hypersurfaces
Cartan Viewpoint and Coordinate Viewpoint
Canonical Spaces of Constant Curvature
Existence
Operators on Forms and Integration
Exterior Derivative
Contraction
Lie Derivative
General Covariant Derivative
Integration of Forms and Stokes' Theorem
Integration in a Riemannian Manifold
Gauss-Bonnet Theory and Rigidity
Gauss-Bonnet Formula
Index Theorem
Gauss-Bonnet Form
Characteristic Forms
Rigidity Problems
Existence Theory
Involutive Distributions and the Frobenius Theorem
The Fundamental Existence Theorem for Hypersurfaces
The Exponential Map
Convex Neighborhoods
Special Coordinate Systems
Isothermal Coordinates and Riemannian Surfaces
Topics in Riemannian Geometry
Jacobi Fields and Conjugate Points
First and Second Variation Formulae
Geometric Interpretation of Riemannian Curvature
The Morse Index Theorem
Completeness
Manifolds with Constant Riemannian Curvature
Manifolds without Conjugate Points
Manifolds with Non-Positive Curvature
Bibliography
Index