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 -bundle. Formally, one can consider the standard Levi-Civita connection for a hypersurface and write the connection:
^4}_X Y = \nabla^M_X Y + h(X,Y)$
where the second fundamental form term acts just like the electromagnetic potential in the established connection-on-principal bundle approach and yields . 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.