Quelques exemples de variétés riemanniennes où toutes les géodésiques issues d'un point sont fermées et de même longueur suivis de quelques résultats sur leur topologie

Bérard-Bergery, Lionel

Annales de l'Institut Fourier, Tome 27 (1977) no. 1, pp. 231-249.

On donne une construction de métriques riemanniennes où toutes les géodésiques issues d’un point sont fermées et de même longueur sur certaines variétés non difféomorphes aux sphères et projectifs usuels, et en particulier sur certaines sphères exotiques

On étudie ensuite la topologie de ces variétés ; on précise le classique théorème de Bott dans le cas non simplement connexe ; on étend ses conclusions (affaiblies) sous une hypothèse plus faible sur les géodésiques.

A construction is given, which puts on certain manifolds, non diffeomorphic to the usual spheres and projective spaces, such Riemannian metrics that all the geodesics from some point are closed and of the same length ; one gets for example some exotic spheres.

Then one studies the topology of such a manifold, refining classical Bott’s theorem in the non-simply connected case ; one gets finally a weaker conclusion under some weaker condition on geodesics.

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