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I would like to ask for a reference to the following statement (hopefully correct):

Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic. Suppose that $\gamma$ is contractible. Then for any contraction of this geodesic at some point its length will be equal to $2\pi$.

It would be even better if there is a reference for the case when $M$ a locally $CAT(1)$ space (not necessarily manifold)

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There is an article by B. Bowditch, Notes on locally CAT(1) spaces, in Geometric Group Theory, R.Charney, M.Davis, and M.Shapiro eds., de Gruyter (1995) that will likely be of help. The article is about curve shortening in locally CAT(1) spaces.

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