Francis Bonahon

MATH 535: Differential Geometry

Fall 09

Instructor: Francis Bonahon
Office: KAP 248G
Telephone: (213) 740-2390
e-mail: fbonahon@math.usc.edu

The course is an introduction to the theory of manifolds and of differential forms. To a large extent, this is a fancy extension of multivariable calculus. It culminates with the theorems of Stokes and de Rham. It forms a natural combination with MATH 540, Topology, which is usually taught in the spring semester. Although these two courses are technically independent of each other and can be taken in any order, many of the concepts introduced in each one of them were historically motivated by material developed in the other.

Here are some of the main topics which will be covered in the course. 

• Submanifolds of Rⁿ 
• Differentiable manifolds
• Tangent vectors
• Submanifolds, immersions, submersions
• The Inverse Function Theorem and applications
• Sard's theorem and application
• Vector fields
  
• The tensor and exterior algebras
• Differential forms
• The exterior derivative; deRham cohomology
• Orientation of manifolds
• Integration of differential forms
• Stokes's Theorem
• de Rham's Theorem
  
• Degree theory
  

Homework assignments

They are posted here.

Notes

I am not planning to closely follow any textbook this semester. Instead, I will try to provide notes which are posted here. If additional help is needed, I can suggest the following books which all have been used as textbooks for the course in the past (and all have one flaw or another):

Marcel Berger, Bernard Gostiaux, Differential geometry: manifolds, curves and surfaces, Springer-Verlag
William Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press
Lawrence Conlon, Differentiable manifolds: a first course, Birkhaüser
Frank Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag
Ib Madsen, Jorgen Tornehave, From calculus to cohomology, Cambridge University Press

The class will meet MWF 11:00 - 11:50 in KAP 141. There will be regular homework assignments, one midterm and one final exam.

Grading scheme:

Homework 50%
Midterm 25%
Final exam 25%