An isometrie $\varphi: S_1\to S_2$ which cannot be extended into distance-preserving map

Question

I'm searching for an example isometrie $\varphi: S_1\to S_2$($S_i$ are regular surfaces) which cannot be extended into distance-preserving maps $F: \Bbb R^3 \to \Bbb R^3$. A reference or hint will help too.

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\color{gray}{No need the proof of isometrie or disability for extending.}

Answer 1

John gave a (pretty standard) example as a comment: the surfaces $$\{(x,y,0):0<x<\pi, 0<y<1\}$$ and $$\{(\cos x,y,\sin x): 0<x<\pi, 0<y<1\}$$ are isometric in their intrinsic metrics, but are not extrinsically isometric.