As mentioned in the title of this question, I want to be able to move from an intrinsic viewpoint of vector bundles, to an extrinsic viewpoint. For manifolds, this would be described via the Nash embedding theorem: $$\text{n-dimensional Riemannian manifold $(M,g)$} \to \text{submanifold of $\mathbb{R}^{2n}$ with canonical metric}.$$
This is incredibly useful as geometric intuition. Using the previous paragraph as context, is there an analogous "Nash embedding theorem" for vector bundles? i.e., can we describe an association $$ \text{rank-$r$ vector bundle $(E,\nabla)$}\to \text{subbundle of $(\underline{\mathbb{R}^{2r}})$ with canonical connection}?$$ As for the definition of canonical, note that every vector bundle admits a realization as a subbundle of a trivial bundle of sufficiently-high rank. The trivial bundle has a canonical connection, and this connection induces a unique connection on the subbundle (just examine the linear algebra of horizontal distributions: a splitting of a vector space induces a splitting of a subspace).

Note, the resulting connection of the subbundle would have to be induced by the parent bundle, analogous to the case of the Nash embedding. The extrinsic viewpoint would allow one to visualize the "Force = curvature" principle in a more concrete setting: hopefully we can make precise what it means to view the evolving connection-form of electromagnetism as the evolving geometry of some vector subbundle.


The counterpart for connections of the Nash embedding is an easier result called the Narasimhan-Ramanan theorem,

M.S. Narasimhan, S. Ramanan: Existence of universal connections, Amer. J. Math., 83(1961), 563-572.

It essentially states that any connection on a vector bundle $E$ is gauge equivalent to a connection induced by the trivial connection on a trivial vector bundle containing $E$ as a subbundle.

As for the statement Force=Curvature I suggest you have a look at Th. Fraenkel's book The Geometry of Physics.

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