# Embedding of a bundle with connection into a bundle with flat connection?

I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection.

Given a smooth manifold $$M$$ together with a vector bundle $$V$$ on $$M$$ equipped with some connection $$D$$, I want to find an orthogonal bundle $$W$$ with a flat connection $$D'$$ such that $$V\subset W$$ is a subbundle and $$D$$ is induced from $$D'$$. Here by "induced" I mean the following: given a section $$s$$ of $$V$$ and a vector field, the covariant derivative of $$s$$ using $$D$$ should be the composition of the covariant derivative of $$s$$ (seen as section of $$W$$) using $$D'$$ with the projection onto $$TV$$ using the orthogonal structure of $$W$$.

A special case is when $$M$$ is Riemannian. For $$V=TM$$ and $$D$$ the Levi-Civita connection, the answer to the question is Nash's embedding theorem: there is an embedding of $$M$$ into some $$\mathbb{R}^N$$ such that the connection $$D$$ is induced by the trivial connection $$d$$ on $$\mathbb{R}^N$$.

Is there a result of this kind? Any references?

To clarify a bit. The tautological bundle over the Grassmannian $$\gamma_k\to Gr_k(\mathbb{R}^N)$$ has a complement $$\gamma^\bot$$ the bundle whose fiber at a subspace $$V$$ is the ortho-complement of $$V$$ in $$\mathbb{R}^N$$. It follows that $$\gamma\oplus\gamma^\bot =Gr_k(\mathbb{R}^N) \times \mathbb{R}^N$$. The connection in this paper is the connection induced from the trivial connection by projection.
If the basis $$M$$ is compact,every bundle has an inverse for the Whitney sum; that is can be embedded into a trivial bundle. Check that it satisfies all the other conditions.
• Thanks for your reply. We can assume $M$ to be compact. But I don't see how the Whitney inverse says anything about the connection. Since the sum gives a trivial bundle we can use the standard derivative. But does it induce the given connection on the initial bundle? Nov 12, 2021 at 9:41