in any complete Riemannian manifold of finite volume is still an open problem (see for instance the following recent review of the subject [@BM19]).

Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------

In Section \[se:preliminaries\], we fix notation and recall results of the variational theory of geodesics that we will need. In Section \[se:Bangert\], we give a comprehensive proof of Theorem \[thm:Bangert\] of Bangert. In Section \[se:contractible\], we prove Theorem \[thm:main\] when hypothesis \[it:contractible\], \[it:selfintersecting\] or \[it:intersecting\] is assumed. In Section \[se:mind\], we prove Theorem \[thm:main\] when hypothesis \[it:mind\] is assumed. In Section \[se:homological\], we prove the last case of Theorem \[thm:main\]. In Section \[se:plane\], we prove Theorem \[thm:plane\].

Acknowledgments {#acknowledgments .unnumbered}
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This work started while the first author was visiting the Ruhr-Universität Bochum (Germany) in October 2019. The first author wishes to thank Alberto Abbondandolo and Stefan Suhr for the invitation, and the Ruhr-Universität Bochum for providing an excellent working environment. The first author is also grateful to his advisor Marco Mazzucchelli who introduced him to the result conjectured by Alberto Abbondandolo. To the latter the second author expresses his gratitude for his continuing help and advice. The second author is partially supported by the SFB/TRR 191 “Symplectic Structures 