When are embeddings into Euclidean space unique up to ambient isometry?

Question

Suppose I have a Riemannian smooth manifold $M$ and a smooth isometric embedding $M \hookrightarrow \mathbb{R}^n$. Is this embedding necessarily unique up to some isometry of $\mathbb{R}^n$? If not, is it possible to insert some extra conditions or structure to make this happen?

Answer 1

To get you started searching, this is known as a rigidity result for the isometric embedding problem.

For surfaces in $\mathbb R^3$ there are some well-known results: Leibmann's theorem gives rigidity for the sphere, and Cohn-Vossen for strictly convex surfaces (i.e. closed surfaces with nonvanishing Gaussian curvature).

The bendability of cylinders rule out a lot of optimistic conjectures. Even for compact surfaces nasty things can happen in general:

(Image pinched from the book of Montriel & Ros via this MO question)

For hypersurfaces there are some general results available under assumptions on the rank of the second fundamental form - see e.g. Vol 5 Chap 12 of Spivak's DG. The introduction of this paper has some good discussion and references, including an older result that gives rigidity for generic metrics if the codimension is low enough.

Answer 2

This certainly isn't true in general. For instance, if $M=\mathbb{R}$, then any smooth injective curve parametrized by arc length is an isometric embedding, but such curves certainly aren't all conjugate up to ambient isometry (if $n>1$). Or if $M$ is disconnected, you can't expect a single ambient isometry to be able to simultaneously do what you want on each of its components. I don't know whether there are any reasonable conditions under which embeddings are unique up to ambient isometry.