Lectures on Closed Geodesics

The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo­ metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l. c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e. , the corres­ ponding periodic orbit in the geodesic flow is infinitesimally stable.

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.