Lectures on Closed Geodesics

1. The Hilbert Manifold of Closed Curves.- 1.1 Hilbert Manifolds.- 1.2 The Manifold of Closed Curves.- 1.3 Riemannian Metric and Energy Integral of the Manifold of Closed Curves.- 1.4 The Condition (C) of Palais and Smale and its Consequences.- 2. The Morse-Lusternik-Schnirelmann Theory on the Manifold of Closed Curves.- 2.1 The Lusternik-Schnirelmann Theory on ?M.- 2.2 The Space of Unparameterized Closed Curves.- 2.3 Closed Geodesics on Spheres.- 2.4 Morse Theory on ?M.- 2.5 The Morse Complex.- 3. The Geodesic Flow.- 3.1 Hamiltonian Systems.- 3.2 The Index Theorem for Closed Geodesics.- 3.3 Properties of the Poincaré Map.- 3.3 Appendix. The Birkhoff-Lewis Fixed Point Theorem. By J. Moser.- 4. On the Existence of Many Closed Geodesics.- 4.1 Critical Points in ?M and the Theorem of Fet.- 4.2 The Theorem of Gromoll-Meyer.- 4.3 The Existence of Infinitely Many Closed Geodesics.- 4.3 Appendix. The Minimal Model for the Rational Homotopy Type of ?M. By J. Sacks.- 4.4 Some Generic Existence Theorems.- 5. Miscellaneous Results.- 5.1 The Theorem of the Three Closed Geodesics.- 5.2 Some Special Manifolds of Elliptic Type.- 5.3 Geodesics on Manifolds of Hyperbolic and Parabolic Type.- Appendix. The Theorem of Lusternik and Schnirelmann.- A.2 Closed Curves without Self-intersections on the 2-sphere.- A.3 The Theorem of Lusternik and Schnirelmann.