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### Contents

Marston Morse and His Mathematical Works by Raoul Bott | vii |

2 A Onetoone Representation of Geodesies on a Surface of Negative Curvature | 1 |

3 Recurrent Geodesies on a Surface of Negative Curvature 21 | 39 |

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admissible admissible set analytic arbitrary boundary C-measure calculus of variations canonical curves closed compact condition conjugate points continuous converges coordinates Corollary corresponding critical points cycle defined definition deformation denote determined diffeomorphism differential distance end points equal equations exists extremal g fc-cap fc-cycle finite number following lemma following theorem follows from Lemma function F given Hence homeomorphism homology homotopic hypothesis infinite integer integral interior interval joining k-caps length linear set manifold mapping Marston Morse Mathematical maximal group meromorphic functions metric metric space Morse Morse theory neighborhood non-degenerate normal p-curve pair parameter partial derivatives periodic extremal positive constant preceding proof prove the following rank recurrent reduced curve relations representation respectively s-function satisfy sense sequence set of points solution space subset sufficiently small suppose surface symbols termed theory topological trajectory transformation type number unit circle variables vector vertex space zero