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Question 1: Given a smooth Riemannian surface $M$ in $R^3$ (i.e., a smooth Riemannian 2-manifold embedded in $R^3$) and a diffeomorphism $f: M\rightarrow M$ of class $C^{k\geq 2}$, does $f$ admit a smooth extension $\tilde{f}$ to all of $R^3$? If not always, then are there sufficient conditions?

Question 2: If the answer to Q1 is affirmative, then given two diffeomorphisms $f,g: M\rightarrow M$ of class $C^{k\geq 2}$ which are close in the $C^2$-topology, can we find extensions $\tilde{f}, \tilde{g}$ which are also close in the $C^2$-topology?

Edit 1: I should add that $M$ carries the induced metric (from $R^3$).

Edit 2: We can ask a more general question. Say $M$ is a smooth Riemannian $m$-manifold. Embed $M$ in $R^N$ isometrically. Say $f: M\rightarrow M$ is a diffeomorphism of class $C^k$. Can we extend $f$ smoothly to $R^N$?

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Question 1: Given a smooth Riemannian surface $M$ in $R^3$ (i.e., a smooth Riemannian 2-manifold embedded in $R^3$) and a diffeomorphism $f: M\rightarrow M$ of class $C^{k\geq 2}$, does $f$ admit a smooth extension $\tilde{f}$ to all of $R^3$? If not always, then are there sufficient conditions?

Question 2: If the answer to Q1 is affirmative, then given two diffeomorphisms $f,g: M\rightarrow M$ of class $C^{k\geq 2}$ which are close in the $C^2$-topology, can we find extensions $\tilde{f}, \tilde{g}$ which are also close in the $C^2$-topology?

I should add that $M$ carries the induced metric (from $R^3$).

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