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Let $M$ be a compact Riemannian manifold and set $\mathcal{M}$ to be the space of $W^{1,2}$ closed curves on $M$. This is a Hilbert manifold and for positive constant $r$ define $\mathcal{M}_E^{r^2}$ to be the set of curves with energy less than or equal to $r^2$ and $\mathcal{M}_L^r$ to be the set of curves with lenght less than or equal to $r$. We know $\mathcal{M}_E^{r^2}\subset\mathcal{M}_L^r$. Is it true in general that the inclusion map $\iota\colon\mathcal{M}_E^{r^2}\to\mathcal{M}_L^r$ is a homotopy equivalence or weak homotopy equivalence. It seems reasonable to me since every closed geodesic in $\mathcal{M}_L^r$ is also contained in $\mathcal{M}_E^{r^2}$ and from the point of view of Morse-Bott theory, the contribution of critical points of energy fucntional to the topology of both of them is the same.

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