# “Nash Style” Embedding Theorem for Connections

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?

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## 1 Answer

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?

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Great. Sorry for inadvertently asking the same question twice. –  Jean Delinez Apr 29 '11 at 14:36
no problem. I even overlooked that it was you who asked the embedding previous question. –  Stefan Waldmann Apr 29 '11 at 14:43
A link to the other question mathoverflow.net/questions/54434/… –  j.c. Apr 29 '11 at 14:46