TITLE: When distance nonincreasing map is an isometry
QUESTION [7 upvotes]: Let $f: M \to M$ be a distance nonincreasing map between a closed Riemannian manifold $M$
and $f$ is homotopic to the idendity map. Is it then $f$ an isometry?

REPLY [11 votes]: Combine the following:

If $X$ is a compact metric space, then any surjective distance non-increasing map $X\to X$ is an isometry (see e.g. Burago-Burago-Ivanov's "A course in metric geometry", theorem 1.6.15, or prove yourself-this is easy). 
If a map of closed manifolds has nonzero $\mathbb Z_2$-degree, the map is surjective (because every point in the target has odd number of preimages).