An isometry is a Diffeomorphism $F: M \to N$ between two Riemannian manifolds with metrics $g_M$ and $g_N$, such that $F^* g_N = g_M$ (where $F^* g_N$ refers to the pull back of the metric, see Riemannian metric for definiton)

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For two Surfaces S1 and S2 if there is a Diffeomorphism $f$ which takes each curve on S1 to a curve on S2 of the same length, then such an $f$ is called an isometry. For this to work, the two surfaces should already be homeomorphic.

Theorem, characterizing isometries by invariance of the First fundamental form.

Examples

Strip of a plane is isometric to cylinder

Cone is isometric to a region of the plane

If a chart is both conformal and area-preserving, then the surface/patch is isometric to the plane. Conformal map, and Area-preserving map are weaker notions of isometry.

How does one prove that there is not an isometric chart between a piece of the plane and a piece of the Sphere? : Theorema Egregium