One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.
The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.
Author(s): Manfredo P. do Carmo
Series: Dover Books on Mathematics
Edition: 2
Publisher: Dover Publications
Year: 2016
Language: English
Pages: 528
City: Mineola, New York
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
