Instructor: Debra Lewis
Office: 359B Baskin Engineering
Phone: 459-2718
E-mail: lewis at ucsc dot edu
(checked more often than voicemail or gmail)
and/or DebraKLewis at gmail dot com
Office hours: M, Th 4:00-5:00
Course web page: http://people.ucsc.edu/~lewis/Math208
(here)
Introduction to Smooth Manifolds, John M. Lee, Springer.
Possible supplemental texts: An Introduction to Differentiable Manifolds and Riemannian Geometry, Boothby. Topology from a Diffential Viewpoint, Milnor.
Scanned supplemental material:
Grassmannians and projective space. Boothby (10/9/08)
Implicit Function Theorem. Marsden and Ratiu (10/9/08)
Local Surjectivity Theorem. Marsden and Ratiu (10/20/08)
Hairy Ball Theorem and Brouwer Fixed Point Theorem. Milnor, American
Math Monthly, July 1978. (10/23/08)
Ruled surfaces Mathematica calculations:
PDF,
notebook. (10/28/08)
Some material on the phase flow and matrix exponential:
The midterm. (11/5/08)
Topics:
Definition of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem, vector fields, flows, and Lie bracket, Frobenius' theorem.
VERY TENTATIVE SCHEDULE
| Tuesday | Thursday |
| September 25: Topological and smooth manifolds | |
| September 30: Smooth functions and maps | October 2: Tangent vectors, derivations, and pushforwards |
| October 7: Coordinate calculations and partitions of unity | October 9: Submersions, immersions, and embeddings |
| October 14: Inverse and Implicit Function Theorems | October 16: Submanifolds, level sets and Lie groups |
| October 21: Vector fields and the tangent bundle | October 23: Integral curves and flows |
| October 28: Fundamental flow theorem, existence and uniqueness | October 30: Lie brackets |
| November 4: VOTE! Distributions and involutivity | November 6: Frobenius' Theorem |
| November 11: HOLIDAY | November 13: Singular points, Sard's Theorem |
| November 18: Whitney Embedding Theorem | November 20: Whitney Approximation Theorem and tubular neighborhoods |
| November 25: Homotopies and approximations | November 27: HOLIDAY |
| December 4: Additional topics/presentations? |
COURSE WORK