## Lectures on Closed GeodesicsThe 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. |

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### Contents

The Hilbert Manifold of Closed Curves | 1 |

12 The Manifold of Closed Curves | 7 |

13 Riemannian Metric and Energy Integral of the Manifold of Closed Curves | 15 |

14 The Condition C of Palais and Smale and its Consequences | 22 |

The MorseLusternikSchnirelmann Theory on the Manifold of Closed Curves | 32 |

22 The Space of Unparameterized Closed Curves | 40 |

23 Closed Geodesics on Spheres | 46 |

24 Morse Theory on AM | 55 |

42 The Theorem of GromollMeyer | 132 |

43 The Existence of Infinitely Many Closed Geodesics | 142 |

The Minimal Model for the Rational Homotopy Type of AM | 156 |

44 Some Generic Existence Theorems | 161 |

Miscellaneous Results | 167 |

52 Some Special Manifolds of Elliptic Type | 177 |

53 Geodesics on Manifolds of Hyperbolic and Parabolic Type | 188 |

The Theorem of Lusternik and Schnirelmann | 203 |

25 The Morse Complex | 65 |

The Geodesic Flow | 77 |

32 The Index Theorem for Closed Geodesics | 86 |

33 Properties of the Poincare Map | 100 |

The BirkhoffLewis Fixed Point Theorem By Jurgen Moser | 115 |

On the Existence of Many Closed Geodesies | 122 |

A1 The Space PM and the Theorem of Lyusternik and Fet | 204 |

A2 Closed Curves without Selfintersections on the 2sphere | 208 |

A3 The Theorem of Lusternik and Schnirelmann | 210 |

219 | |

225 | |

### Common terms and phrases

Anosov type assume canonical choose circle claim closed curves coefficients cohomology compact Riemannian manifold completes the proof conjugate points consider coordinates critical point cycle define deformation denote dense diffeomorphism differentiable manifold dimension eigenvalue element equation equivariant exists fibre finite follows function fundamental group geodesic flow geodesic segment given grad Hence Hilbert manifold homology classes hyperbolic implies induced integer invariant isometric isomorphism Jacobi field Klingenberg Lemma length Math metric g minimizing geodesic Moreover Morse complex Morse theory multiplicity non-degenerate critical submanifold non-trivial normal bundle Note obtain open neighborhood orthogonal parameterized periodic orbit prime closed geodesics Proposition prove representation respectively restriction Riemannian manifold Riemannian metric S-action S-orbits satisfying self-intersections sequence simply connected sphere strong unstable manifold submanifold subset subspace sufficiently small symmetric spaces symplectic transformation tangent bundle Theorem topology twist type unparameterized unstable manifolds vector field Wuu(c zero