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 Geodesics

8.1 Riemannian metrics and distance
8.2 Length minimizing curves
8.3 The existence of geodesics
8.4 First examples
8.5 Geodesics on $SO(n)$
8.6 Geodesics on $U(n)$ and $SU(n)$
8.7 Geodesics and matrix groups

Appendix

Appendix 8.1: The proof of the vector field theorem

9 Properties of Geodesics

9.1 The maximal extension of a geodesic
9.2 The exponential map
9.3 Gaussian coordinates
9.4 The proof of the geodesic theorem

10 Principal Bundles

10.1 The definition
10.2 A cocycle definition
10.3 Principal bundles constructed from vector bundles
10.4 Quotients of Lie groups by subgroups
10.5 Examples of Lie group quotients
10.6 Cocycle construction examples
10.7 Pull-backs of principal bundles
10.8 Reducible principal bundles
10.9 Associated vector bundles

Appendix

Appendix 10.1: Proof of Proposition 10.1

11 Covariant Derivatives and Connections

11.1 Covariant derivatives
11.2 The space of covariant derivatives
11.3 Another construction of covariant derivatives
11.4 Principal bundles and connections
11.5 Connections and covariant derivatives
11.6 Horizontal lifts
11.7 An application to the classification of principal $G$-bundles up to isomorphism
11.8 Connections, covariant derivatives and pull-back bundles

12 Covariant Derivatives, Connections, and Curvature

12.1 Exterior derivative
12.2 Closed forms, exact forms, diffeomorphisms, and De Rham cohomology
12.3 Lie derivative
12.4 Curvature and covariant derivatives
12.5 An example
12.6 Curvature and commutators
12.7 Connections and curvature
12.8 The horizontal subbundle revisited

13 Flat Connections and Holonomy

13.1 Flat connections
13.2 Flat connections on bundles over the circle
13.3 Foliations
13.4 Automorphisms of a principal bundle
13.5 The fundamental group
13.6 The flat connections on bundles over $M$
13.7 The universal covering space
13.8 Holonomy and curvature
13.9 Proof of the classification theorem for flat connections

Appendices

Appendix 13.1: Smoothing maps
Appendix 13.2: The proof of the Frobenius theorem

14 Curvature Polynomials and Characteristic Classes

14.1 The Bianchi Identity
14.2 Characteristic forms
14.3 Characteristic classes: Part 1
14.4 Characteristic classes: Part 2
14.5 Characteristic classes for complex vector bundles and the Chern classes
14.6 Characteristic classes for real vector bundles and the Pontryagin classes
14.7 Examples of bundles with nonzero Chern classes
14.8 The degree of the map $g \to g^m$ from $SU(2)$ to itself
14.9 A Chern–Simons form

Appendices

Appendix 14.1: The ad-invariant functions on $M(n; \mathbb{C})$
Appendix 14.2: Integration on manifolds
Appendix 14.3: The degree of a map