Cornell Math - MATH 652, Fall 2004

MATH 652: Differentiable Manifolds I (Fall 2004)

Instructor: Reyer Sjamaar

Meeting Time & Room

Prerequisites: undergraduate analysis, linear algebra and point-set topology.

This course is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle and a section of the tangent bundle is a vector field. Alternatively vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric we will develop the notions of parallel transport, curvature and geodesics. We examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits we will give a brief introduction to de Rham cohomology, Lie groups, or other optional topics.