Differential Geometry I: MAT 4930 Section 2E15
Fall 2014
MWF 8th period (3:00-3:50), Little 203

Instructor: David Groisser
groisser@ufl.edu

Course summary. This course will cover what I would usually cover in MTG 6256, the first semester of a year-long graduate sequence introduces the tools of differential geometry and differential topology. It is being run with the 4000 special-topics number to give very advanced undergraduates inexpensive access to a course with this content. The course is intended only for undergraduates who have the preparation and maturity for a real 6000-level course.

Since MTG 6256 is always followed by MTG 6257 in the spring, parts of the description below assume that our class will continue with a second semester in the spring. Whether that actually happens will depend on student interest.

The fall semester will be devoted primarily to the foundations of manifold theory. Topics will include a brief review of advanced calculus from a geometric viewpoint; definition and examples of manifolds maps of manifolds; critical points and the Regular Value Theorem; vector fields, flows, and Lie derivatives; exterior algebra and differential forms; integration on oriented manifolds; Stokes theorem ; vector bundles and tensor bundles (possibly deferred to spring); and possibly an introduction to Riemannian metrics and Riemannian geometry. Riemannian geometry is a large topic; most of this material will probably be deferred to the second semester. There is a wide variety of topics that could be covered in the second semester, so topics in the second semester will depend upon students’ interests. Please see prerequisites below.

Likely topics for the spring semester include: introduction to Riemannian geometry (if not done in the first semester); surfaces in R³ and the Gauss-Bonnet theorem; connections on principal bundles and associated vector bundles. Some possibilities for additional topics for the spring semester are:

Further study in Riemannian geometry (conjugate points on geodesics,
Hopf-Rinow Theorem, Morse index, curvature-comparison theorems, …)

Lie groups and Lie algebras

Symplectic geometry and the geometry of classical mechanics

Complex and Kaehler manifolds

Selected topics in differential topology (transversality,
Poincare-Hopf Theorem, degree theory, embedding theorems, …)

There will be time only for a very limited number of these topics (some of them are semester-long topics by themselves). Student input will be sought before a final decision is made.

Prerequisites: MAA 4212 and its whole prerequisite-chain.

Pre- or co-requisite: MAS 4301.