**Basic Differential Geometry (Spring Semester) – 60670**

- Connections in vector bundles: Covariant derivative, parallel transport, orientability, curvature, baby Chern-Weil.
- Riemannian geometry: Levi Cevita connection, exponential map, Jacobi fields, arc length variation formulas, fundamental equations for metric immersions and submersions, space forms, Hopf-Rinow, Hadamard-Cartan, Bonnet- Myers (Gauss-Bonnet, Bochner technique).
- Other geometric structures: (one or more of Kahler manifolds, symplectic manifolds, contact manifolds).

*References*

Chavel, *Riemannian Geometry: A modern introduction.*

Grove, *Riemannian geometry: A metric entrance.*

Petersen, *Riemannian Geometry.*

Gallot, Hulin, and Lafontaine, *Riemannian geometry.*

Cullen, *Introduction to General Topology.*

Dugundji, *Topology.*

Kelley, *General **Topology. *Munkres, *Topology.** *

*S*teen, *Counterexamples in Topology.*