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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?

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3 Answers 3

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The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmannian is universal for connections not just bundles. That is any connection in a U(n) or O(n) bundle is pulled back from the canonical connection in the appropriate Grassmannian by a map. Since the canonical connection has the desired form so does so does the original connection. They also estimate the required rank of the complement.

To clarify a bit. The tautological bundle over the Grassmannian $\gamma_k\to \operatorname{Gr}_k(\mathbb{R}^N)$ has a complement $\gamma_k^\perp$ the bundle whose fiber at a subspace $V$ is the ortho-complement of $V$ in $\mathbb{R}^N$. It follows that $\gamma_k\oplus\gamma_k^\perp =\operatorname{Gr}_k(\mathbb{R}^N) \times \mathbb{R}^N$. The connection in this paper is the connection induced from the trivial connection by projection.

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  • $\begingroup$ Great, thanks for the reference! There is a second paper from the same authors dealing with the case of non-compact structure groups. Is it clear that the pull-back of the canonical connection is the same as the "induction process" described in my question? $\endgroup$
    – AThomas
    Commented Nov 12, 2021 at 9:58
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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.

https://en.wikipedia.org/wiki/Inverse_bundle

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    $\begingroup$ 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? $\endgroup$
    – AThomas
    Commented Nov 12, 2021 at 9:41
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I think the paper "Inverses of structured vector bundles" has exactly what you need.

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