Differential Geometry of Curves and Surfaces

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 R^3
	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
Index