Math 5041 - Geometry I

Fall 2010

Section information: Classes meet on Tuesdays and Thursdays from 1:00PM to 2:30PM, in Cupples I room 111.

Subject: Here is the course catalogue's description of math 5041: An introductory graduate level course in geometry/topology including differential calculus in n-space, vector fields and flows, differential forms and calculus on manifolds; elements of Lie groups and Lie algebras; Frobenius theorem; elements of Riemannian geometry. I'll try not to deviate too much from this description. More details on the content below.

Text: Introduction to smooth manifolds by John M. Lee. (Springer, Graduate texts in mathematics, 2003.)

Tentative list of topics:

• Smooth manifolds, maps and vector fields;
• The tangent and cotangent bundles, vector bundles;
• Submersions, immersions, embeddings, submanifolds;
• Lie groups, Lie algebras;
• Tensor fields, differential forms;
• Integration on manifolds, Stokes theorem;
• Foliations, Frobenius theorem.

Coursework: Grades will be based on homework assignments, a mid-term exam and a final. (Exams will be take-home.) I expect that we will have a total of approximately 12 assignments throughtout the semester. They will be due on the Thursday of each week.