دانلود کتاب Differential Geometry of Curves and Surfaces

فهرست مطالب :

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