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.