Basic Differential Geometry

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.