Courses / Module

Module Objective: to develop advanced skills in geometry with an emphasis on Riemannian geometry as well as topics in pseudo-Riemannian geometry relating to theoretical physics. This will ensure students receive a thorough grounding in topics like smooth manifold theory, tensor analysis and curvature.

Core Topics: smooth manifolds and corresponding tangent and vector bundles, tensors including differential forms and Riemannian/pseudo-Riemannian metrics, connections, geodesics and curvature (sectional, Ricci and scalar), exponential map, extrinsic curvature, second fundamental form and the Gauss curvature equation.

Optional Extra Topics (which may vary) drawn from: Jacobi fields, comparison geometry, curvature constraints and topological obstructions, Gauss-Bonnet Theorem.