Instructor: Ko Honda
Office: KAP 406E
E-mail: kohonda at usc dot edu with the first o removed
Telephone: 213-740-3785
URL: http://www-bcf.usc.edu/~khonda
Office Hours: Mondays and Wednesdays, 1-2 pm
Grader: Russell Avdek

Topics

1. Review of advanced calculus (calculus on R^n); inverse and implicit function theorems.
2. Differentiable manifolds and their maps.
3. Differential forms: tensor and exterior algebra, exterior differentiation, and Lie derivatives.
4. Integration: Stokes' theorem, de Rham cohomology, and computations using Meyer-Vietoris sequences.
5. The tangent bundle: vector fields, distributions, Frobenius' theorem.
6. Riemannian geometry: metrics, connections, and curvature.

• Warner, Foundations of Differentiable Manifolds and Lie Groups
  

• Course Notes

• Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry.
• Bott & Tu, Differential Forms in Algebraic Topology: Chapter 1, Sections 1.1-1.5.
• Guillemin & Pollack, Differential Topology: Chapters 1,4.
• Morita, Geometry of Differential Forms. (This reference is probably closest in spirit to this course.)

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, 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, which can be found here.  Homework is due on Mondays, 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

There will be a take-home final.  This will be approximately 30% of your final grade.