Differential Geometry: Manifolds, Curves, and Surfaces Manifolds, Curves, and Surfaces
This book consists of two parts, different in form but similar in spirit. The first, which comprises chapters 0 through 9, is a revised and somewhat enlarged version of the 1972 book Geometrie Differentielle. The second part, chapters 10 and 11, is an attempt to remedy the notorious absence in the o...
Main Authors:  , 

Format:  eBook 
Language:  English 
Published: 
New York, NY
Springer New York
1988, 1988

Edition:  1st ed. 1988 
Series:  Graduate Texts in Mathematics

Subjects:  
Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 0. Background
 0.0 Notation and Recap
 0.1 Exterior Algebra
 0.2 Differential Calculus
 0.3 Differential Forms
 0.4 Integration
 0.5 Exercises
 1. Differential Equations
 1.1 Generalities
 1.2 Equations with Constant Coefficients. Existence of Local Solutions
 1.3 Global Uniqueness and Global Flows
 1.4 Time and ParameterDependent Vector Fields
 1.5 TimeDependent Vector Fields: Uniqueness And Global Flow
 1.6 Cultural Digression
 2. Differentiable Manifolds
 2.1 Submanifolds of Rn
 2.2 Abstract Manifolds
 2.3 Differentiable Maps
 2.4 Covering Maps and Quotients
 2.5 Tangent Spaces
 2.6 Submanifolds, Immersions, Submersions and Embeddings
 2.7 Normal Bundles and Tubular Neighborhoods
 2.8 Exercises
 3. Partitions of Unity, Densities and Curves
 3.1 Embeddings of Compact Manifolds
 3.2 Partitions of Unity
 3.3 Densities
 3.4 Classification of Connected OneDimensional Manifolds
 3.5 Vector Fields and Differential Equations on Manifolds
 3.6 Exercises
 4. Critical Points
 4.1 Definitions and Examples
 4.2 NonDegenerate Critical Points
 4.3 Sard’s Theorem
 4.4 Exercises
 5. Differential Forms
 5.1 The Bundle ?rT*X
 5.2 Differential Forms on a Manifold
 5.3 Volume Forms and Orientation
 5.4 De Rham Groups
 5.5 Lie Derivatives
 5.6 Starshaped Sets and Poincaré’s Lemma
 5.7 De Rham Groups of Spheres and Projective Spaces
 5.8 De Rham Groups of Tori
 5.9 Exercises
 6. Integration of Differential Forms
 6.1 Integrating Forms of Maximal Degree
 6.2 Stokes’ Theorem
 6.3 First Applications of Stokes’ Theorem
 6.4 Canonical Volume Forms
 6.5 Volume of a Submanifold of Euclidean Space
 6.6 Canonical Density on a Submanifold of Euclidean Space
 6.7 Volume of Tubes I
 6.8 Volume of Tubes II
 6.9 Volume of Tubes III.6.10 Exercises
 7. Degree Theory
 7.1 Preliminary Lemmas
 7.2 Calculation of Rd(X)
 10.6 What the Second Fundamental Form Is Good For
 10.7 Links Between the two Fundamental Forms
 10.8 A Word about Hypersurfaces in Rn+1
 11. A Guide to the Global Theory of Surfaces
 11.1 Shortest Paths
 11.2 Surfaces of Constant Curvature
 11.3 The Two Variation Formulas
 11.4 Shortest Paths and the Injectivity Radius
 11.5 Manifolds with Curvature Bounded Below
 11.6 Manifolds with Curvature Bounded Above
 11.7 The GaussBonnet and Hopf Formulas
 11.8 The Isoperimetric Inequality on Surfaces
 11.9 Closed Geodesics and Isosystolic Inequalities
 11.10 Surfaces AU of Whose Geodesics Are Closed
 11.11 Transition: Embedding and Immersion Problems
 11.12 Surfaces of Zero Curvature
 11.13 Surfaces of NonNegative Curvature.11.14 Uniqueness and Rigidity Results
 11.15 Surfaces of Negative Curvature
 11.16 Minimal Surfaces
 11.17 Surfaces of Constant Mean Curvature, or Soap Bubbles
 11.18 Weingarten Surfaces
 11.19 Envelopes of Families of Planes
 11.20 Isoperimetric Inequalities for Surfaces
 11.21 A Potpourri of Characteristic Properties
 Index of Symbols and Notations
 7.3 The Degree of a Map
 7.4 Invariance under Homotopy. Applications
 7.5 Volume of Tubes and the GaussBonnet Formula
 7.6 SelfMaps of the Circle
 7.7 Index of Vector Fields on Abstract Manifolds
 7.8 Exercises
 8. Curves: The Local Theory
 8.0 Introduction
 8.1 Definitions
 8.2 Affine Invariants: Tangent, Osculating Plan, Concavity
 8.3 Arclength
 8.4 Curvature
 8.5 Signed Curvature of a Plane Curve
 8.6 Torsion of ThreeDimensional Curves
 8.7 Exercises
 9. Plane Curves: The Global Theory
 9.1 Definitions
 9.2 Jordan’s Theorem
 9.3 The Isoperimetric Inequality
 9.4 The Turning Number
 9.5 The Turning Tangent Theorem
 9.6 Global Convexity
 9.7 The FourVertex Theorem
 9.8 The FabriciusBjerreHalpern Formula
 9.9 Exercises
 10. A Guide to the Local Theory of Surfaces in R3
 10.1 Definitions
 10.2 Examples
 10.3 The Two Fundamental Forms
 10.4 What the First Fundamental Form Is Good For
 10.5 Gaussian Curvature