Math 652 — Fall 2001 Differentiable Manifolds

This is an introduction to differential geometry at the level of a beginning graduate student. This course serves as a good introduction to students aiming to take further courses in Lie groups, algebraic geometry, topology, dynamical systems and analysis on manifolds.

Prerequisites: advanced calculus, linear algebra (Mathematics 431), point set topology (Mathematics 453).

From the course catalog: Topological manifolds, smooth manifolds, immersions and embeddings, Tangent bundles, fiber bundles, Vector fields and dynamical systems, Frobenius' theorem, Lie groups, Integration on manifolds, differential forms, Stokes theorem. Connections, Riemannian manifolds, geodesics, Curvature, Gauss-Bonnet theorem, Tubular neighborhoods, transversality and cobordism.

Other topics (as time permits):
Sheaves and de Rham's theorem
Hodge theory.

Textbooks:
W. Boothby, An introduction to differentiable manifolds and Riemannian geometry
S. Helgason, Differential geometry and symmetric spaces
S. Kobayashi-K. Nomizu, Foundations of differential geometry
M. Spivak, Differential geometry
F. Warner, Foundations of differentiable manifolds and Lie groups