It is quite well-known that Maxwell’s equations can be expressed via differential forms and there are many presentations for such. So flux densities for electric and magnetic fields are 2-forms; the current density is a 2-form. The electric and magnetic field intensities are 1-forms and the Maxwell’s equations are:

These equations have perfect sense on . The last two equations say that the flux densities are closed forms which implies, because the sphere has no cohomology in dimension 2 that in fact both are exact. This note is simply to observe that electromagnetism on a 4-sphere can be understood extremely simply on the entire 4-sphere. I will come back to the issue of how it is described via differential forms for a principal -bundle in terms of a connection and then use the description of connections via a 1-forms on the total space of the bundle and then note that for a hypersurface of a sphere, the analogue of the circle-bundle occurs from the geodesic normal circles and the connection 1-form has the analogue of a 1-form on the 4-sphere. This latter issue should have testable implications in the real universe because it would imply that actual electromagnetism in our universe does NOT have full gauge invariance.

The standard presentation for the principal bundle approach is to use the electric potential and the magnetic potential to form, on the trivial circle bundle over space-time the differential 1-form

so that the Maxwell’s equations become where . There is no difficulty in producing such a 1-form for where is a coordinate along the normal direction. However, since the four-spere is not the total space of a principal circle bundle, one cannot use the first coordinate globally.

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