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I have a question about the existence of geodesics on a compact Riemannian manifold $M$. Is there an elementary way to prove that in each nontrivial free homotopy class of loops, there is a closed geodesic $\gamma$ on $M$?

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Let be $[\gamma]$ nontrivial free homotopy class of loops and $l=\inf_{\beta; \beta\in[\gamma]}l(\beta)$ where $l(\beta)$ is a lenght of the curve $\beta.$ We will show that there is a geodesic $\beta$ in $[\gamma]$ such that $l(\beta)=l.$ Let be $\beta_n$ a sequence of loops in $[\gamma]$ such that $l(\beta_n)\to l.$

The first intuition is that the sequence $\beta_n$ converges to the desired curve, but this is not quite true.

We can assume without loss of generality that each $\beta_n$ is a geodesic by parts and are parameterized by arc length.

Let us show that beta has a subsequence that converges uniformly to a continuous loop $\beta.$ In fact, as the curves $ \beta_n $ are parameterized by arc length, we have $$ d(\beta_j(t_1),\beta_j(t_2))=|t_1-t_2| $$

Therefore the set $\{\beta_n\}$ is a uniformly limited and equicontínuos set, how $M$ is compact follows from the Arzelá-Ascoli theorem that there exists a subsequence $\beta_{n_j}$ that converges uniformly for a continuous loop $\beta_0:[0,1]\to M$

Now let $t_0<t_1\cdots<t_n $ be a finite partition of $[0,1]$ such that each $\beta_0([t_i,t_{i+1}])$ is contained in a totally normal neighborhood.

Now consider the geodesic by parts $\beta:[0,1]\to M$ such that in each $[t_i,t_{i+1}]$ the curve $\beta$ is equal to geodesic segment connecting the points $\beta_0(t_i)$ e $\beta_0(t_{i+1})$

A contradiction argument shows that $l(\beta)=l$

An argument of shortcuts shows that $\beta$ is a geodesic and minimizing

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Do you know of a particular example where the $\beta_n$s themselves don't converge, so that taking the subsequence is necessary? –  Jason DeVito Jan 5 '13 at 16:06
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@Jason: Take an embedded torus. Fix two shortest loops $C_1, C_2$ and squeeze the torus to make its circumference smaller along $C_1$ and $C_2$. (All of this while keeping the torus the same outside of a neighborhood of $C_1$ and $C_2$.) Then $C_1$ and $C_2$ both minimize the length in their homotopy class in the squeezed torus. Take $\beta_n = C_1$ if $n$ is even, and $\beta_n = C_2$ if $n$ is odd. In this example you need to choose a subsequence. –  Sam Jan 5 '13 at 16:23
    
@SamL.: Thanks! That's so obvious I can't believe I didn't see it myself! –  Jason DeVito Jan 5 '13 at 16:25
    
@JasonDeVito: You're welcome! –  Sam Jan 5 '13 at 16:29

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