Course Announcement

Differential Geometry I and II : MTG 6256-6257
Fall 2012 - Spring 2013
Fall time and room: MWF 7th period (1:55-2:45), Little 217

Instructor: David Groisser (

Course summary. This year-long graduate sequence introduces the tools of differential geometry and differential topology.

MTG 6256, the first semester of the sequence, 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 (including Lie groups and submanifolds of Euclidean space); maps of manifolds; critical points and the Regular Value Theorem; Sard's Theorem (possibly deferred to spring); vector fields, flows, and Lie derivatives; exterior algebra and differential forms; Stokes' theorem and de Rham cohomology; vector bundles and tensor bundles (possibly deferred to spring); Riemannian metrics; introduction to Riemannian geometry; and connections and curvature. Riemannian geometry and the theory of connections are large topics; most of this material will probably be deferred to the second semester, MTG 6257. Please see prerequisites for MTG 6256 below. Here is a (tentative) syllabus.

MTG 6257. If time forbids covering the following topics in the fall semester (MTG 6256), they will likely be covered in the spring (MTG 6257): surfaces in R3 and the Gauss-Bonnet theorem; connections on principal bundles and associated vector bundles. (STUDENTS IN OTHER DEPARTMENTS TAKE NOTE: Even if the material on surfaces in R3 is deferred to the spring semester, you will not be allowed to take MTG 6257 in the spring unless you take MTG 6256 in the fall. If the only differential geometry topics that interest you are surfaces in R3 and other potential spring topics, you still need to take MTG 6256 in the fall. No exceptions.) 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

Elliptic PDE on manifolds and Hodge Theory

Curvature and characteristic classes

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 for MTG 6256

Essentially, these can be summarized as "Everything you would learn as an undergraduate (pure-)math major". Differential geometry pulls together strands of all three major divisions of mathematics: analysis, algebra, and topology. Here are some particulars:

Required textbook for MTG 6256

   See syllabus.

Some reference books

Last update made by D. Groisser Sun Aug 5 19:36:23 EDT 2012