Mth 624 Advanced Differential Geometry I

Topics selected from differentiable manifolds, differential forms, DeRham cohomology, Lie groups, fibre bundles, the Riemannian metric, affine and Riemannian connections, parallel translations, holonomy, geodesics, curvature, isometric embeddings and hypersurfaces, the Second Fundamental Form, complete Riemannian manifolds and the Hopf-Rinow theorem, spaces of constant curvature, variations of arc length, and the Morse Index theorem. This is the first course in a sequence of three: Mth 624, Mth 625, and Mth 626. Expected preparation: Mth 425/Mth 525.