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 (groisser@math.edu)

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:

• Linear algebra, including inner-product spaces.
  This is essential. By linear algebra I don't just mean matrix algebra (although the basics of matrix algebra are still essential) or numerical linear algebra (working with large linear systems on a computer); I mean the general concepts and theory of vector spaces, linear transformations, and inner products. Differential geometry uses linear algebra probably more than any other field does, and it's important that you be comfortable with it. By the end of the course you will know a lot more about linear algebra than when you started, but you have to start at a reasonable level. If you are not clear on the difference between a matrix and a linear transformation, or if you have trouble working with a vector space that is not given to you explicitly as a subspace of Rⁿ, you are not ready to learn differential geometry. For those of you familiar with UF's linear algebra courses, a course similar to MAS 4105 would be a sufficient prerequisite (provided you did well and have retained what you learned); a course similar to MAS 3114 or MAD 6406 would not be sufficient.

• Advanced calculus. Although the topics below will be reviewed briefly, students entering the class should already be familiar with:
  • the derivative viewed as a linear transformation
  • the chain rule
  • the Inverse Function Theorem and Implicit Function Theorem
  • the fundamental existence/uniqueness theorem for (systems of) ordinary differential equations

• Basic point-set topology
  • metric spaces (familiarity with more general topological spaces would be helpful as well)
  • open and closed sets; compactness; connectedness; completeness
  • continuous maps and homeomorphisms
  These topics are reviewed briefly in this Point-Set Topology: Glossary and Review, which students should go through before the course starts.

• Algebra: basics of groups, rings, homomorphisms, and quotient constructions. I hesitate to call all of this a prerequisite, since much of it will be used only occasionally. However, when it is used, if you're unfamiliar with the foundational material I'm using, it will be up to you to learn what you need on your own.

Required textbook for MTG 6256

   See syllabus.

Some reference books