By the well-known Whitney embedding theorem, any manifold can be embedded in [itex]\mathbb R^n[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# In how many ways can you embed a Riemannian manifold in R^n?

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