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).

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.
• Steen, Counterexamples in Topology.