Math 545/546: Topology and Geometry of Manifolds
John M. Lee
Winter/Spring 2000, MWF 1:30-2:20
This course continues the study of manifolds begun in Math 544.
For these two quarters, the subject will be smooth or differentiable
manifolds, which are manifolds on which derivatives of functions and
maps make sense. We will study the basic flora and fauna that live
on them: submanifolds, tangent vectors, vector fields, flows, Riemannian
metrics and their simple properties, tensor fields, differential forms,
orientations. The basic theory and examples of Lie groups (which are groups
that are also manifolds) will be woven throughout the course.
Text: Introduction to Smooth Manifolds, notes by the instructor;
these will be handed out week by week during the course.
Prerequisites: Math 544 and the following undergraduate topics:
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Vector calculus: Calculus of mappings from Rn
to Rm, partial derivatives, the chain rule, the
total derivative as a linear map, multiple integrals, the change of variables
formula, the inverse function theorem, divergence, gradient, curl, and
the theorems of Green, Gauss, and Stokes.
-
Ordinary differential equations: basic theory and techniques at
the level of Math 307 and 309.
-
Linear algebra: abstract vector spaces, dimension, linear maps,
change of basis, Jordan canonical form, inner products, bilinear forms,
linear functionals, dual spaces.
Homework and grading: Grades will be based on weekly problem sets
and a final exam.