Course overview
The geometry of Euclid, flat space geometry, was believed for a long time to be the only geometry possible. In the 19th century examples were produced of curved space geometries, by various authors. Of these, the most far-reaching and general was developed by B. Riemann. This course is a modern introduction to Riemanns geometry, starting from the notion of smooth manifold, Riemannian metric, connections, geodesics, and curvature. This proved the ideal setting for Einsteins study of relativity, for instance.
- Smooth manifolds, vector bundles and connections
- Riemannian metrics and curvature
- Geodesics, distance, and completeness
Course learning outcomes
- Demonstrate an understanding of the intrinsic geometry of manifolds
- Apply the general theory to specific examples
- Produce sophisticated arguments and proofs, at the appropriate level
- Communicate results clearly