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## Interpolation between S4 and principal U(1) bundles

Suppose $M$ is a three dimensional submanifold of $S4(1/h)$.  For each point x in M, there is a unique normal circle starting and returning to $x$ with exact length $2 \pi/h$.  In the exceptional case that none of these geodesics intersect each other, we can use the standard machinery of connection forms and curvatures to define electromagnetism.  General theory says the potential is a connection 1-form on the bundle that at each point chooses the vertical component of the tangent space whose kernel is a horizontal subspace.  The curvature of such a connection is the field strength for electromagnetism or Yang-Mills fields.  But in the non-exceptional case, the geodesics intersect each other and on the points of intersection ambiguities arise for the horizontal subspaces.

One possible approach to resolution of this problem is to consider the map $\Phi: M \times S^1 -> S^4(1/h)$ which maps $(x,t)$ to the point traversed following the normal geodesic from $x$ for time $t/h$.  The useful thing about this map is that it resolves the ambiguities and one can define the usual sort of connection 1-form and curvature on the initial space.  This map does not resolve the fundamental problem however as the issue of whether there exists a one-form on S4 is just hidden.

As a preliminary step to understanding electromagnetism in the actual universe, one might consider the problem of determining the connection one-forms on $M\times S^1$ which are pull-backs of one-forms on $S^4(1/h)$.  Both first and second cohomologies of $S4(1/h)$ are trivial, so every closed 2-form is exact.  The pull-backs also have the same properties.  In particular the Maxwell’s equations in the form $dF = 0$ for $F = \Phi^*(G)$ will have a potential one-form which is a pull-back of a one-form as well.