Original French edition published by Presses Universitaires de France -PUF
1988, XII, 476 p.
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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 original book of any treatment of surfaces in three-space, an omission all the more unforgivable in that surfaces are some of the most common geometrical objects, not only in mathematics but in many branches of physics. Geometrie Differentielle was based on a course I taught in Paris in 1969- 70 and again in 1970-71. In designing this course I was decisively influ enced by a conversation with Serge Lang, and I let myself be guided by three general ideas. First, to avoid making the statement and proof of Stokes' formula the climax of the course and running out of time before any of its applications could be discussed. Second, to illustrate each new notion with non-trivial examples, as soon as possible after its introduc tion. And finally, to familiarize geometry-oriented students with analysis and analysis-oriented students with geometry, at least in what concerns manifolds.
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 Parameter-Dependent Vector Fields.- 1.5 Time-Dependent 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 One-Dimensional Manifolds.- 3.5 Vector Fields and Differential Equations on Manifolds.- 3.6 Exercises.- 4. Critical Points.- 4.1 Definitions and Examples.- 4.2 Non-Degenerate 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 Star-shaped 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).- 7.3 The Degree of a Map.- 7.4 Invariance under Homotopy. Applications.- 7.5 Volume of Tubes and the Gauss-Bonnet Formula.- 7.6 Self-Maps 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 Three-Dimensional 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 Four-Vertex Theorem.- 9.8 The Fabricius-Bjerre-Halpern 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.- 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 Gauss-Bonnet 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 Non-Negative 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 Pot-pourri of Characteristic Properties.- Index of Symbols and Notations.