Instructor: Ko Honda
Office: MS 7901
Office Hours: Wed 9-10 am, 11-12 noon
E-mail: honda at math dot ucla dot edu.
Telephone: 310-825-2143
URL: http://www.math.ucla.edu/~honda

TA: Michael Miller; office hours TuTh 1-2pm; smmiller at ucla dot edu

Topics

1. Review of advanced calculus (calculus on R^n); inverse and implicit function theorems.
2. Differentiable manifolds and their maps.
3. Tangent and cotangent bundles, vector bundles.
4. Vector fields and integral curves.
5. Sard's theorem, transversality, Whitney embedding theorem.
6. Oriented intersection theory: degree, Lefschetz fixed point theorem.
   

Prerequisites

• Knowledge of calculus on R^n, as presented in the first three chapters of Spivak's Calculus on Manifolds book.
• This course requires more mathematical maturity than the average first-year graduate course in the mathematics department.  I will not follow any text closely in the first half of the course, and it is up to you to look up the necessary information from the various references.  (Of course, I would be happy to suggest where to look up things.)
  

There will be weekly problem sets; see the class schedule.  Homework is due on Fridays, although there may be some exceptional weeks.  The problem sets count for a large percentage of your total grade (approximately 70%).  You may work with others or consult other textbooks, but the homework you turn in must be written by you, in your own words, and you must cite your sources used and your collaborators!

Final examination

1. Differential Geometry Course Notes
   
2. Guillemin & Pollack, Differential Topology.
   
3. Spivak, A Comprehensive Introduction to Differential Geometry
4. Tu, An Introduction to Manifolds
   
5. Warner, Foundations of Differentiable Manifolds and Lie Groups
6. Peter Petersen's notes