Last modified: 1 February 2018
MATH 740 Differentiable manifolds: transversality, inverse and implicit function theorems, immersions and submersions, submanifolds. Fiber bundles, basics of Lie groups, vector bundles, tangent and cotangent bundles. Differential forms, Lie derivative, integration on manifolds and Stokes' theorem. Riemannian metrics, connections, curvature, covariant differentiation. Minimizing properties of geodesics, Hopf-Rinow theorem, Jacobi equations. REFERENCES for MATH 740: M. DoCarmo, "Riemannian Geometry" F. Warner, "Foundations of Differentiable Manifolds and Lie Groups" J. Lee, "Introduction to Smooth Manifolds" B. O'Neill, "Semi-Riemannian Geometry" P. Petersen, "Riemannian Geometry" J. Milnor, "Topology from the Differentiable Viewpoint"