Math 652 — Differentiable Manifolds I

Fall 2002

Instructor: Reyer Sjamaar

Time: MW 8:40-9:55

Room: Malott 205

Differentiable manifolds are a kind of topological spaces on which one can do differential calculus. They appear in many areas of mathematics and its applications, such as for example Riemannian and symplectic geometry, topology, dynamical systems, partial differential equations, representation theory and mathematical physics. This course is an introduction to manifolds at the level of a beginning graduate student. Prerequisites are advanced calculus, linear algebra (Math 431) and point-set topology (Math 453). Here is an outline:

Basic notions: manifolds, embeddings, partitions of unity etc.

Structures on manifolds: tangent vectors, tensors.

Vector fields and their flows (with applications to ordinary and partial differential equations).

Tensor calculus (various ways of differentiating functions and tensors).

Lie groups and homogeneous spaces (an important source of examples).

There should be some time for additional topics, such as differential forms and De Rham's theorem, or metrics and curvature, or transversality and Morse theory. (Suggestions are welcome.)

There will probably not be a required textbook. Material will be culled from the following and other sources:

Boothby, An Introduction to differentiable manifolds

Warner, Foundations of differentiable manifolds and Lie groups

Spivak, A Comprehensive Introduction to Differential geometry, vol. 1