Math 623: Differential Geometry 2

Instructor: Dr. Robles
Lecture: Blocker 624, MWF 9:10 - 10:00 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.

Prerequisites. Math 622 and Math 636 (or familiarity with the notions topological space and homeomorphism).

Reference text. Foundations of Differential Geometry, vol. 1, by Kobayashi & Nomizu.

Topics.

  1. Riemannian manifolds (review).
  2. de Rham cohomology: the Laplacian and the Hodge Theorem.
  3. Vector bundles.
  4. Principle bundles.
  5. Covariant differentiation.
  6. Curvature.
  7. Holonomy and the de Rham decomposition.
  8. Geodesics and the exponential map: the Hopf-Rinow Theorem.

Other references.

All three texts (Kobayashi--Nomizu, Warner and Madsen--Tornehave) are on 4 hour reserve at the library.

Due         Homework
Sep 11 HW: 7.1, 7.2; 9.10 -- 9.14.
Sep 18 HW: 9.9, 9.15, 9.16; 6.1, 6.2, 6.3.
Oct 02 G-structures: 1 -- 10.
Oct 09 G-structures 2: 9 -- 11, 14, 16.
Oct 16 Parallel Transport.
Oct 23 Geodesics: 9.15, 9.16.
Oct 30 HW.
Nov 06 HW.
Nov 13 HW: 1, 3, 4, 6.
Nov 20 HW.

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