It is a stupid question i guess but like they say if you ask you are stupid for 5 minutes and if you don't ask you are stupid forever . here is the question given a closed manifold (M,g) $(M,g)$ and \alpha $\alpha$ in \pi_1(M,p) $\pi_1(M,p)$ we can define the length of \alpha $\alpha$ as the minimum riemannian length of a representative now obviousle length \alpha^2 $\alpha^2$ is less or equal then 2xlength(\alpha) $2\mathrm{length}(\alpha)$ but it seems it is always equal 2xlength(\alpha) $2\mathrm{length}(\alpha)$ i want to know why ?
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riemannian length of an element of the fundamental group of a manifoldIt is a stupid question i guess but like they say if you ask you are stupid for 5 minutes and if you don't ask you are stupid forever . here is the question given a closed manifold (M,g) and \alpha in \pi_1(M,p) we can define the length of \alpha as the minimum riemannian length of a representative now obviousle length \alpha^2 is less or equal then 2xlength(\alpha) but it seems it is always equal 2xlength(\alpha) i want to know why ?
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