Foundational material; De Rahm cohomology and harmonic differential form; parallel transport, connections, and covariant derivatives; geodesics and Jacobi fields; morse theory and closed geodesics; symmetric spaces and Kohler manifolds; the Palais-Smale condition and closed geodesics; harmonic maps; variational problems from quantum field theory. Appendix: linear elliptic partial differential equations; fundamental groups and covering spaces.
