Author | Berger, Marcel. author |
---|---|
Title | Differential Geometry: Manifolds, Curves, and Surfaces [electronic resource] / by Marcel Berger, Bernard Gostiaux |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1033-7 |
Descript | XII, 476 p. online resource |
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 Sardโs 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 Poincarรฉโs 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 Stokesโ Theorem -- 6.3 First Applications of Stokesโ 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 Jordanโs 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