Core topics in differential and Riemannian geometry including Lie groups, curvature, relations with topology.
Text at the level of Riemannian Geometry of do Carmo's or Gallot-Hulin-Lafontaine.
- Riemannian metrics and geodesics
- Examples of Riemannian manifolds (submanifolds, submersions, warped products, homogeneous spaces, Lie groups)
- Covariant derivative, parallel transport
- Geodesics, exponential map, Hopf-Rinow
- Curvature tensor, Ricci, sectional, mean, scalar curvatures, spaces of constant curvature, curvature computaions for examples listed in 2).
- 1st and 2nd variation formulas, Jacobi fields, Rauch and Ricatti comparison, and applications such as Myers and Cartan-Hadamard theorems
- Selections from more advanced topics such as: volume comparision and Ricci curvature, minimal surfaces, spectral geometry, Hodge theory, symmetric spaces and holonomy, comparison geometry and Lorentz geometry