There are two notions of isometry between Riemannian 2-manifolds:
- a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and
- a "metric-preserving" map $f$ with $I(x) = I(f(x))$
($I(x)$ being the first fundamental form)
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?