Daniel Ruberman
Fall 2005

Syllabus for Math 110a: Geometric Analysis

1. Manifolds, change of coordinates, differential structure.
2. Tangent bundle, derivations, vector fields, Lie bracket, tensors.
3. Basics of vector bundles: normal bundles, pullback construction.
4. Transversality and implicit function theorems.
5. Picard theorem and Frobenius Theorem.
6. Differential forms, closed and exact, Poincaré Lemma. Frobenius Theorem in differential form version.
7. Integration, Stokes Theorem, orientatations and volume elements, de Rham cohomology and theorem.

Optional (if time permits): Basic Lie Groups - Lie algebra, one parameter subgroups, structural equations, left and right invariant vector fields.

Text:

M Spivak. A Comprehensive Introduction to Differential Geometry, vol. I.

Additional References:

F. Warner, Foundations of Differentiable Manifolds and Lie groups.
J. Milnor. Topology from the Differentiable Viewpoint.
R. Bott and L. Tu, Differential Forms in Algebraic Topology.