Math 8365 Spring 2025

Contents (tentative)

1 Smooth Manifolds

1.1 Smooth manifolds
1.2 The inverse function theorem and implicit function theorem
1.3 Submanifolds of $\mathbb{R}^m$
1.4 Submanifolds of manifolds
1.5 More constructions of manifolds
1.6 More smooth manifolds: The Grassmannians

Appendices

Appendix 1.1: How to prove the inverse function and implicit function theorems
Appendix 1.2: Partitions of unity

2 Matrices and Lie Groups

2.1 The general linear group
2.2 Lie groups
2.3 Examples of Lie groups
2.4 Some complex Lie groups
2.5 The groups $SL(n; \mathbb{C})$, $U(n)$, and $SU(n)$
2.6 Notation with regards to matrices and differentials

Appendix

Appendix 2.1: The transition functions for the Grassmannians

3 Introduction to Vector Bundles

3.1 The definition
3.2 The standard definition
3.3 The first examples of vector bundles
3.4 The tangent bundle
3.5 Tangent bundle examples
3.6 The cotangent bundle
3.7 Bundle homomorphisms
3.8 Sections of vector bundles
3.9 Sections of $TM$ and $T^*M$

4 Algebra of Vector Bundles

4.1 Subbundles
4.2 Quotient bundles
4.3 The dual bundle
4.4 Bundles of homomorphisms
4.5 Tensor product bundles
4.6 The direct sum
4.7 Tensor powers

5 Maps and Vector Bundles

5.1 The pull-back construction
5.2 Pull-backs and Grassmannians
5.3 Pull-back of differential forms and push-forward of vector fields
5.4 Invariant forms and vector fields on Lie groups
5.5 The exponential map on a matrix group
5.6 The exponential map and right/left invariance on $GL(n; \mathbb{C})$ and its subgroups
5.7 Immersion, submersion and transversality

6 Vector Bundles with $\mathbb{C}^n$ as Fiber

6.1 Definitions
6.2 Comparing definitions
6.3 Examples: The complexification
6.4 Complex bundles over surfaces in $\mathbb{R}^3$
6.5 The tangent bundle to a surface in $\mathbb{R}^3$
6.6 Bundles over 4-dimensional submanifolds in $\mathbb{R}^5$
6.7 Complex bundles over 4-dimensional manifolds
6.8 Complex Grassmannians
6.9 The exterior product construction
6.10 Algebraic operations
6.11 Pull-back

7 Metrics on Vector Bundles

7.1 Metrics and transition functions for real vector bundles
7.2 Metrics and transition functions for complex vector bundles
7.3 Metrics, algebra and maps
7.4 Metrics on $TM$

8 Principal Bundles

8.1 The definition
8.2 A cocycle definition
8.3 Principal bundles constructed from vector bundles
8.4 Quotients of Lie groups by subgroups
8.5 Examples of Lie group quotients
8.6 Cocycle construction examples
8.7 Pull-backs of principal bundles
8.8 Reducible principal bundles
8.9 Associated vector bundles

Appendix

Appendix 8.1: Proof of Proposition 8.1

9 Some Riemannian Geometry

9.1 Covariant derivatives on TM
9.2 Riemannian Curvatures
9.3 Geodesics

10 Covariant Derivatives and Connections

10.1 Covariant derivatives
10.2 The space of covariant derivatives
10.3 Another construction of covariant derivatives
10.4 Principal bundles and connections
10.5 Connections and covariant derivatives
10.6 Horizontal lifts
10.7 An application to the classification of principal $G$-bundles up to isomorphism
10.8 Connections, covariant derivatives and pull-back bundles

11 Covariant Derivatives, Connections, and Curvature

11.1 Exterior derivative
11.2 Closed forms, exact forms, diffeomorphisms, and De Rham cohomology
11.3 Lie derivative
11.4 Curvature and covariant derivatives
11.5 An example
11.6 Curvature and commutators
11.7 Connections and curvature
11.8 The horizontal subbundle revisited

12 Flat Connections and Holonomy

12.1 Flat connections
12.2 Flat connections on bundles over the circle
12.3 Foliations
12.4 Automorphisms of a principal bundle
12.5 The fundamental group
12.6 The flat connections on bundles over $M$
12.7 The universal covering space
12.8 Holonomy and curvature
12.9 Proof of the classification theorem for flat connections

Appendices

Appendix 12.1: Smoothing maps
Appendix 12.2: The proof of the Frobenius theorem

13 Curvature Polynomials and Characteristic Classes

13.1 The Bianchi Identity
13.2 Characteristic forms
13.3 Characteristic classes: Part 1
13.4 Characteristic classes: Part 2
13.5 Characteristic classes for complex vector bundles and the Chern classes
13.6 Characteristic classes for real vector bundles and the Pontryagin classes
13.7 Examples of bundles with nonzero Chern classes
13.8 The degree of the map $g \to g^m$ from $SU(2)$ to itself
13.9 A Chern–Simons form

Appendices

Appendix 13.1: The ad-invariant functions on $M(n; \mathbb{C})$
Appendix 13.2: Integration on manifolds
Appendix 13.3: The degree of a map