The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) 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 Levi-Civita 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? 



"Nash Style" Embedding Theorem for Connections
The standard connection is the Levi-Civita 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 Levi-Civita connection of a Riemannian metric on your manifold. Thus the question is whether a given connection is the Levi-Civita of some Riemannian metric. That was one of your earlier questions. Does that work?


