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Let $(M;g)$ be a closed manifold . fix a point $x$ in $M$ and denote by $G=\pi_1(M;x)$ now let $\alpha$ in G i have 2 questions :

1- can $\alpha$ be represented by a closed geodesic ?

2- can $\alpha $ be represented by a closed minimizing geodesic ?

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    2 can't be true if the loop in $\pi_1$ can only be represented by a self-intersecting curve, which happens frequently.2011-09-26

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