By the well-known Whitney embedding theorem, any manifold can be embedded in [itex]\mathbb R^n[/itex]. You might have also heard the Nash embedding theorem, which basically says that this is still true for Riemannian manifolds (i.e. now we demand the metric is induced from [itex]\mathbb R^n[/itex]). So fine, any Riemannian manifold can be seen as a submanifold of [itex]\mathbb R^n[/itex]. But my question is: in how many ways can one do this? For example, embedding a sphere in [itex]\mathbb R^3[/itex], it doesn't matter where we do it (translation symmetry) or how we orient it (rotational symmetry). I.e. the only freedom in embedding it is the symmetry of [itex]\mathbb R^3[/itex]. So an alternative formulation of the question is: are there isometric submanifolds of [itex]\mathbb R^n[/itex] which are not the same up to rotation and translation?