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.