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?
Best Answer
The paper “Existence of universal connections” by Narasimhan, M. S.; Ramanan, S. proves that the Grassmanian 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 grassmanian 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 $\gamma_k\to Gr_k(\mathbb{R}^N)$ 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.