Differential Geometry: Manifolds Curves and Surfaces

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. Language: English