1.Fundations material. 2.De Rham Cohomology and Harmonic differential forms. 3.Parallel transport, connections, and covariant derivatives. 4.Geodesics and Jacobi fields. 5.More theory and closed geodesics. 6.Symmetric spaces and Kahler manifols. 7.The palais-Smale conditions and closed geodesics. 8.Harmonic maps. 9.Variational problems from quantum field theory.