This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

### Modern Geometry I

I Differential Manifolds

- Smooth manifolds
- Tangent spaces, tangent bundles
- Smooth maps (submersions, immersions, embeddings)
- Vector fields (flows, Frobenius’s theorem)
- Lie groups and homogeneous spaces
- Tensors, differential forms, Stokes’s theorem

II Riemannian Geometry

- Riemannian metrics and connections
- Geodesics
- Curvature
- Jacobi fields
- Isometric immersions (second fundamental form)
- Hopf-Rinow and Cartan-Hadamard theorems
- Manifolds of constant curvature
- Bonnet-Myers theorem

### Modern Geometry II

I Differential Topology

- Transversality
- Tubular neighborhoods
- Intersection theory (mod 2 and oriented)
- Degrees
- Poincare-Hopf index theorem
- Lefschetz fixed-point theorem
- de Rham cohomology
- Poincare duality

II Vector Bundles and Principal Bundles

- Real and complex vector bundles
- Metrics, connections, and curvature on vector bundles
- Chern, Pontryagin, and Euler classes
- Principal bundles
- Connections and curvature on principal bundles
- Parallel transport and holonomy