The strong Whitney embedding theorem states that any smooth (Hausdorff and secondcountable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian manifold could be embedded into Euclidean space in such a way that the metric of the manifold would coincide with the standard dot product. This means that the LeviCivita connection for the manifold maps to the standard connection. More generally, for a manifold $M$ with connection $\nabla$, when does there exist an embedding into Euclidean space such that the connection is mapped to the standard connection?
1 Answer
The standard connection is the LeviCivita connection of the flat metric. So if you have an embedding such that the given connection is the (projection of the) flat connection then you can also induce a metric on the embedded manifold. Hence the given metric was already a LeviCivita connection of a Riemannian metric on your manifold. Thus the question is whether a given connection is the LeviCivita of some Riemannian metric. That was one of your earlier questions. Does that work?

$\begingroup$ Great. Sorry for inadvertently asking the same question twice. $\endgroup$ Apr 29, 2011 at 14:36

$\begingroup$ no problem. I even overlooked that it was you who asked the embedding previous question. $\endgroup$ Apr 29, 2011 at 14:43

$\begingroup$ A link to the other question mathoverflow.net/questions/54434/… $\endgroup$– j.c.Apr 29, 2011 at 14:46