El. knyga: Differential Geometry: Manifolds, Curves, and Surfaces: 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
Sards 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 Poincarés 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
Jordans 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.