Isometric manifolds can be mapped by a motion?

Question

Does the following result hold ?

Let $M$ and $N$ two smooth submanifolds of $(\mathbb R^n, g_{\mathrm{can}})$ such that $M$ and $N$ are isometric as smooth Riemannian manifolds. Then there exists an isometry $f : \mathbb R^n \to \mathbb R^n$ such that $f(M)=N$.