I have an intuition which doesn't mean is correct. Suppose we have $\mathbb{R}^n$ seen as a riemannian manifold with it's standard riemannian metric (i.e. the euclidean one).
Suppose we have a submanifold of $\mathbb{R}^n$ (without boundary) call it $M$ we can define a riemannian metric $g$ by inheriting it from $\mathbb{R}^n$.
What I wonder now is if we fix the Riemannian metric is it possible to uniquely define the submanifold?
I think an answer might be yes, but I wouldn't know how to prove or disprove it.
Any reference you can suggest?