Intro
Contents
Preface
1. Manifolds
1.1 Preliminaries
1.1.1 Space and Coordinatization
1.1.2 The implicit function theorem
1.2 Smooth manifolds
1.2.1 Basic definitions
1.2.2 Partitions of unity
1.2.3 Examples
1.2.4 How many manifolds are there?
2. Natural Constructions on Manifolds
2.1 The tangent bundle
2.1.1 Tangent spaces
2.1.2 The tangent bundle
2.1.3 Transversality
2.1.4 Vector bundles
2.1.5 Some examples of vector bundles
2.2 A linear algebra interlude
2.2.1 Tensor products
2.2.2 Symmetric and skew-symmetric tensors 3.3.6 Connections on tangent bundles
3.4 Integration on manifolds
3.4.1 Integration of 1-densities
3.4.2 Orientability and integration of differential forms
3.4.3 Stokes' formula
3.4.4 Representations and characters of compact Lie groups
3.4.5 Fibered calculus
4. Riemannian Geometry
4.1 Metric properties
4.1.1 Definitions and examples
4.1.2 The Levi-Civita connection
4.1.3 The exponential map and normal coordinates
4.1.4 The length minimizing property of geodesics
4.1.5 Calculus on Riemann manifolds
4.2 The Riemann curvature 4.2.1 Definitions and properties
4.2.2 Examples
4.2.3 Cartan's moving frame method
4.2.4 The geometry of submanifolds
4.2.5 Correlators and their geometry
4.2.6 The Gauss-Bonnet theorem for oriented surfaces
5. Elements of the Calculus of Variations
5.1 The least action principle
5.1.1 The 1-dimensional Euler-Lagrange equations
5.1.2 Noether's conservation principle
5.2 The variational theory of geodesics
5.2.1 Variational formulæ
5.2.2 Jacobi fields
5.2.3 The Hamilton-Jacobi equations
6. The Fundamental Group and Covering Spaces 6.1 The fundamental group
6.1.1 Basic notions
6.1.2 Of categories and functors
6.2 Covering Spaces
6.2.1 Definitions and examples
6.2.2 Unique lifting property
6.2.3 Homotopy lifting property
6.2.4 On the existence of lifts
6.2.5 The universal cover and the fundamental group
7. Cohomology
7.1 DeRham cohomology
7.1.1 Speculations around the Poincaré lemma
7.1.2 Čech vs. DeRham
7.1.3 Very little homological algebra
7.1.4 Functorial properties of the DeRham cohomology
7.1.5 Some simple examples
7.1.6 The Mayer-Vietoris principle