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A great article on Weyl’s development of electromagnetism is here:  varadarajan-weyl-em.  He introduced the notion of an electromagnetic field being the curvature of a connection on a $U(1)$-bundle.  Formally, one can consider the standard Levi-Civita connection for a hypersurface $M\subset S^4(1/h)$ and write the connection:
$\nabla^{$^4}_X Y = \nabla^M_X Y + h(X,Y)\$
where the second fundamental form term $A=h(X,Y)$ acts just like the electromagnetic potential in the established connection-on-principal bundle approach and yields $F_{\mu\nu} = A_{\mu;\nu} - A_{\nu;\mu} + [A_{\mu},A_{\nu}]$.  In subkmanifold geometry, this is a known issue (see CriticalPointTheory Chapter 2).  Thus formally, the principal bundle approach is identical to the normal connection approach where the vector potential is simply the shape operator.