Differential Geometry of Curves and Surfaces

Cover
Title Page
Copyright Page
Dedication Page
Table of Contents
Preface to the Second Edition
Preface
Some Remarks on Using this Book
1. Curves
	1-1 Introduction
	1-2 Parametrized Curves
	1-3 Regular Curves; Arc Length
	1-4 The Vector Product in R3
	1-5 The Local Theory of Curves Parametrized by Arc Length
	1-6 The Local Canonical Form
	1-7 Global Properties of Plane Curves
2. Regular Surfaces
	2-1 Introduction
	2-2 Regular Surfaces; Inverse Images of Regular Values
	2-3 Change of Parameters; Differentiable Functions on Surface
	2-4 The Tangent Plane; The Differential of a Map
	2-5 The First Fundamental Form; Area
	2-6 Orientation of Surfaces
	2-7 A Characterization of Compact Orientable Surfaces
	2-8 A Geometric Definition of Area
	Appendix: A Brief Review of Continuity and Differentiability
3. The Geometry of the Gauss Map
	3-1 Introduction
	3-2 The Definition of the Gauss Map and Its Fundamental Properties
	3-3 The Gauss Map in Local Coordinates
	3-4 Vector Fields
	3-5 Ruled Surfaces and Minimal Surfaces
	Appendix: Self-Adjoint Linear Maps and Quadratic Forms
4. The Intrinsic Geometry of Surfaces
	4-1 Introduction
	4-2 Isometries; Conformal Maps
	4-3 The Gauss Theorem and the Equations of Compatibility
	4-4 Parallel Transport. Geodesics.
	4-5 The Gauss-Bonnet Theorem and Its Applications
	4-6 The Exponential Map. Geodesic Polar Coordinates
	4-7 Further Properties of Geodesics; Convex Neighborhoods
	Appendix: Proofs of the Fundamental Theorems of the Local Theory of Curves and Surfaces
5. Global Differential Geometry
	5-1 Introduction
	5-2 The Rigidity of the Sphere
	5-3 Complete Surfaces. Theorem of Hopf-Rinow
	5-4 First and Second Variations of Arc Length; Bonnet’s Theorem
	5-5 Jacobi Fields and Conjugate Points
	5-6 Covering Spaces; The Theorems of Hadamard
	5-7 Global Theorems for Curves: The Fary-Milnor Theorem
	5-8 Surfaces of Zero Gaussian Curvature
	5-9 Jacobi’s Theorems
	5-10 Abstract Surfaces; Further Generalizations
	5-11 Hilbert’s Theorem
	Appendix: Point-Set Topology of Euclidean Spaces
Bibliography and Comments
Hints and Answers