In general doing any explicit computation for hypersurfaces of a four-sphere is intractable. In 1968 Jim Simons proved that the Laplacian of the squared length of the second fundamental form has a formula for minimal hypersurfaces, i.e. hypersurfaces with zero mean curvature; this was generalized for manifolds with constant mean curvature by Nomizu-Smyth in 1969. This generalizes to, with the second fundamental form and the mean curvature this lemma of G. Huisken:
Suppose is a coframe adapted to with and being the dual of the unit normal vector field. The Maxwell’s equation is
This is just a simple picture. The idea is that electromagnerism, considered a gauge theory is considered with a Lagrangian that is invariant under a gauge transformation via a connection on a principal -bundle where the action of along fibers preserves the Lagrangian. I believe that in our actual universe, there is no principal -bundle but rather the actual universe is a scaled 4-sphere where the substitute for true action is translation in the normal direction. Parallel translation of the frame along the normal geodesic along is the substitute for the circle action. The standard geometric interpretation of is that it is the curvature of a connection of a -bundle; sections of this bundle represent charged fields.
A simple idea is to consider the normal covector as the electromagnetic potential. So . Trivially . Now . In a simple situation where the coframe diagonalizes the symmetric second fundamental form globally we just get and the Maxwell’s equations are trivial. In general, we have the nontrivial condition . Smooth diagonalization of is made possible by results of SmoothDiagonalization
If the physical universe is a compact hypersurface of then how could electromagnetism work? In the established treatment of electromagnetism, e.g. here), where is a -connection so the electromagnetic field is invariant under . Now if the physical universe is so there is a normal circle of length at each point in the physical universe, then a simple first question is how to describe the analogue of the sections of the -bundle which now are deformations of in . So suppose is a smooth map.
The ‘topological’ part of Maxwell’s equations, is clearly fine so long as the deformation is a diffeomorphism since the exterior derivative commutes with pullbacks:
So the problematic part is to understand where the asterisk is the Hodge operator.
It is useful to know the image of tangent vectors by a deformation of to . The map is . For a tangent vector one finds the Jacobi field with initial condition and the image in of is our interest. These Jacobi fields are known explicitly for constant sectional curvatures (JacobiFields): in terms of a tangent frame in their components will be