Basic Differential Geometry (Spring Semester) – 60670

  1. Connections in vector bundles: Covariant derivative, parallel transport, orientability, curvature, baby Chern-Weil.
  2. 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).
  3. Other geometric structures: (one or more of Kahler manifolds, symplectic manifolds, contact manifolds).

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.
Steen, Counterexamples in Topology.