# LECTURE NOTES

##
Geometry and Topology II (Math 528)

Spring 2005, PSU

- Lecture Notes 1

Review of basics of Euclidean Geometry, Cauchy-Schwarz inequality, and basic topology.

- Lecture Notes 2

Definition of manifolds and some examples.

- Lecture Notes 3

Immersions and Embeddings. Proof of the embeddibility of comapct manifolds in Euclidean space.

- Lecture Notes 4

Definition of differential structures and smooth mappings between manifolds.

- Lecture Notes 5

Definition of Tangent space. Characterization of tangent space as derivations of the germs of functions. Differential map and diffeomorphisms.

- Lecture Notes 6

Proofs of the inverse function theorem and the rank theorem.

- Lecture Notes 7

Smooth submanifolds, and immersions. Proof of the smooth embeddibility of smooth manifolds in Euclidean space. Tangent Bundles.

- Lecture Notes 8

Proof of Whitney's 2n+1 embedding theorem.

- Lecture Notes 9

Regular values, proof of fundamental theorem of algebra, Smooth manifolds with boundary, Sard's theorem, and proof of Brouwer's fixed point theorem.

- Lecture Notes 10

Proof of Sard's theorem.

**Lecture Notes 11**: *Chapters 5 and 6 of Spivak, Vol. I.*

Existence and uniqueness of flows, Lie bracket, Lie derivative, the rectifiability theorem for vector fields, and the the Frobenius theorem.

**Lecture Notes 12**: *Chapter 4 of Guillemin and Pollack*

Differential forms and Stokes theorem.