**Daniel Ruberman
**

- Manifolds, change of coordinates, differential structure.
- Tangent bundle, derivations, vector fields, Lie bracket, tensors.
- Basics of vector bundles: normal bundles, pullback construction.
- Transversality and implicit function theorems.
- Picard theorem and Frobenius Theorem.
- Differential forms, closed and exact, Poincaré Lemma. Frobenius Theorem in differential form version.
- 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.*