# Math 102: Foundations of Smooth Manifolds

Fall 2011

 Instructor Craig J. Sutton Lectures MWF 12:30-1:35 (201 Kemeny) X-Hour Tu 1-1:50 (201 Kemeny) Office Hours Tu & F 10:30-Noon (Also By Appointment) Office 321 Kemeny Hall E-mail Craig.J.Sutton AT You Know Where Phone 603-646-1059 Syllabus/Homework Click here

Course Description:

Manifolds provide mathematicians and other scientists with a way of grappling with the concept of "space." The space occupied by an object. The space that we inhabit. Or, perhaps, the space of configurations of a mechanical system. While manifolds are center stage in the study of geometry and topology, they also provide an appropriate framework in which to explore aspects of mathematical physics, dynamics, control theory, medical imaging, econometrics and robotics, to name just a few. This course will serve as an introduction to the basics of manifold theory and lay the foundations needed to explore problems in which "space" plays a fundamental role.

Topics will include some of the following

• manifolds & tangent bundles
• vector fields & vector bundles
• smooth maps and the inverse function theorem
• cotangent bundle & differential forms
• densities, integration on manifolds and the generalized theorem of Stokes
• Whitney's imbedding theorem
• Lie groups, group representations and group actions
• homogeneous spaces
• Tensor Fields & Riemannian metrics
• Differential forms & orientation
• integral curves & the existence and uniqueness theorem for ODEs
• Distributions and Frobenius' Theorem

Prerequisites: Linear Algebra (Math 24), point-set topology (Math 54) and multivariable analysis (Math 73). It will also help to be familiar with covering spaces and the fundamental group, but the interested student should be able to pick this up on the fly.

Course Objectives: We will provde you with the tools needed to deal with spaces as they arise in the "wild" and on the qualifying exam in differential topology.

Textbook: An Introduction to Differentiable Manifolds and Riemannian Geometry (Revised 2nd Edition), William M. Boothby, Academic Press 2003. (available at Wheelock Books).

Deliverables & (tentative) Grading Guide: The following will probably comprise the written assignemtns for this term.

• (Quasi) Weekly Homework: There will be (quasi) weekly homework assignments for this class which will generally be assigned on Wednesday and collected the follwiong Wednesday. No late homework will be accepted.
• Oral Final Exam (Closed Book): More details later
• Graduate Students who have passed the qualifying exam in topology can enroll in this course and will be exempt from all assignments.

Your course grade will probably be computed as follows.

 Homework 70 % Final Exam 30%

Students with disabilities: If you have a disability and require disability related accomodations please speak to me and Ward Newmeyer, Director of Student Accessibility Services, as soon as possible so we can find a remedy.

Last Updated 21 September, 2011