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Hi, J. Ge, the $2$ to $1$ map from $S^1$ to itself is not a "Riemannian" covering map if the metric on $S^1$ is fixed. I agree with you, it is neither distance non-increasing, nor distance non-decreasing. But if we use two different metrics on $S^1$ such that this map is a Riemannian covering map (just pull back the metric), then it is distance non-increasing. Then it is not a local isometry from $S^1$ to itself, of course. That is what Misha means?
Why if $f$ is not an isometry, then $Vol (f(M)< vol (M)$?For example, a covering map is distance nonincreasing and $Vol f(M)=vol (M)$, but a covering map is not an isometry. Of course a covering map is not homotopic to the idendity map.
