Math 622: Differential Geometry 1

Homework.

Midterms
    1 : Solutions.
    2 : Wednesday, March 25.
    3 : Solutions.

Final May 11, 8-10 am.

E-mail. I will be contacting the class though the TAMU e-mail system. Be sure to check your neo.tamu.edu account regularly, or have those e-mails forwarded to your preferred account. I've already sent the class an e-mail. Contact me if you did not receive it.

Prerequisites. Linear algebra (eg. 323), vector calculus (eg. 221) and enough topology to be comfortable with then notions topological space and homeomorphism. If you are short on the prerequisites, or would prefer to study classical surface theory, then you might consider Math 439.

Refrence text. A Comprehensive Introduction to Differential Geometry, Volume 1, by Michael Spivak.

Topics.

1. Smooth manifolds and mappings.
   1. Inverse and implicit function theorems.
   2. Immersions, embeddings, submersions, the rank theorem.
   3. Surfaces in 3-space; generalizations to submanifolds of Euclidean space.
   4. Lie groups, Lie algebras and the Maurer-Cartan form.
   5. Grassmannians and homogeneous spaces.
2. Differential forms.
   1. Tensor calculus.
   2. Frobenius Theorem. (Cartan's Theorem.)
   3. Integration on manifolds.
   4. Stokes's Theorem.
   5. Poincare Lemma and deRham cohomology.
3. Riemannian manifolds
   1. Gauss and mean curvature on surfaces, Gauss-Bonnet theorem.
   2. Riemannian curvature.

Other references.

• Frank Morgan's Riemannian Geometry: A Beginner's Guide is a slim, very accessible text, and will provide a nice introduction. It is perhaps a little short on content, but does cover the basics and is a good place for a geometry neophyte to begin. (Morgan's book would make good reading during Winter Break.)
• If you'd like some more substantial reading, try out Frank W. Warner's Foundations of Differentiable Manifolds and Lie Groups.
• A nice readable text that falls between the two is James R. Munkres's Analysis on Manifolds.

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