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I'm reading the proof of the Preissman Theorem, in Do Carmo's book of Riemannian Geometry. A crucial step in this demonstration is the following lema,

Lema: Let $M$ be a compact riemannian manifold, and $\alpha$ a non trivial deck transformation of the universal covering $\widetilde{M}$, where we are considering $\widetilde{M}$ with a covering metric. So the statement is that $\alpha$ leaves invariant a geodesic $\widetilde{\gamma}$ of $\widetilde{M}$, in this sense

$$\alpha(\widetilde{\gamma}(-\infty,\infty))=\widetilde{\gamma}(-\infty,\infty).$$

Sketch of proof: Let $\pi:\widetilde{M}\to M$ be the covering transformation. Let $\widetilde{p}\in \widetilde{M}$ and $p=\pi(\widetilde{p}).$ Let $g\in \pi_1(M,p)$ be the element corresponding to $\alpha$ by the known isomorphism $\pi_1(M,p)\simeq Aut(\widetilde{M}).$ By the Cartan Theorem, there is a closed geodesic $\gamma$ in the class of free homotopy $M$ given by $g.$

The main idea now is to show that, $\alpha$ fixes the extension of a lifting of $\gamma.$ For this, we obtain a deck tranformation that clearly fix the lifting of $\gamma$ (just take a deck transformation $\beta$ associated to the class of homotopy of $\gamma$ with a base point $q\in \gamma$). And then show that they coincide in one point and therefore must be the same, $\alpha=\beta$.

My Question: Is there any reason to believe that the geodesic wich will be fixed by $\alpha$ is precisely the lifting of a geodesic given by the Cartan Theorem? Or was that just an insight wich the person who'd demonstrated the theorem have?

For those who do not remember this is the statement of the theorem cartan

Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed curve which is geodesic in all of its points.)

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I am ashamed, because the answer seems simple. Given a non trival element, $g\in \pi_1$, it is clear that the deck transfomation $\alpha$ associated to $g$ leaves invariant the extension of a lifting of any element of the class $g.$ Thus, the more natural curves associated with our purpose are the liftings of the elements of the class $g.$ However, there are not necessarily geodesics in the class of $g$ (See this post Cartan Theorem.). So, by Cartan Theorem, there exists a closed geodesic in the free class determined by $g$. And as $\alpha$ leaves invariant the liftings of the elements of the class $g$, is natural to ask if $\alpha$ leaves invariant the lifting of this geodesic.

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