Pseudo-Riemannian geodesic foliations by circles

Pierre Mounoud (1) , Stefan Suhr (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Fachbereich Mathematik [Hamburg]

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an $S^1$-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by $S^1\times \R$. It follows that every Lorentzian surface contains a non-closed geodesic.

Domaines: Géométrie différentielle [math.DG]

Pierre Mounoud, Stefan Suhr. Pseudo-Riemannian geodesic foliations by circles. Mathematische Zeitschrift, 2013, 274 (1-2), pp.225--238. ⟨10.1007/s00209-012-1066-0⟩. ⟨hal-00654794v3⟩