Different notions of isometry for Riemannian 2-manifolds

There are two notions of isometry between Riemannian 2-manifolds:

The second isometry surely implies the first, but what about the other direction? Can there be metrics and isometries (in the first sense) that don't imply isometries in the second sense?

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I suppose you assume that $f$ is one to one. Because if you don't take any covering map $f : X \rightarrow Y$ and pull back the first fundamental form on the covering map X. Then the second condition is true while $X$ and $Y$ can't be isometric –  Selim Ghazouani Feb 27 '13 at 8:33
I think this answers your question: en.wikipedia.org/wiki/Myers-Steenrod_theorem –  muzzlator Feb 27 '13 at 8:35
Someone did an online write-up discussing this theorem here: amathew.wordpress.com/2009/11/17/… The wiki page also references this paper of Palais where he talks about recovering the entire structure of the Riemannian manifold purely from knowing the metric. ams.org/journals/proc/1957-008-04/S0002-9939-1957-0088000-X/… –  muzzlator Feb 27 '13 at 9:00
@muzzlator "Someone" is Akhil Mathew, an active user and a former moderator of Math.SE. –  user53153 Mar 4 '13 at 22:14