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.
Last modified:
March 25, 2004
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