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## On unification of gravity and electromagnetism on S4

I have held the conviction that the right way to unify gravity and electromagnetism in S4 is by literally identifying the second fundamental form terms in the Ricci curvature equation for hypersurfaces of a 4-sphere with the stress energy tensor of electromagnetism.  This leads to $F_{ab} = g(\nabla_n e_a, e_b)$ where $n$ is normal to the hypersurface.  On an S4, we have the very special situation that for every hypersurface, the normal line forms a circle; but it is not the total space of any $U(1)$-bundle.  In a coframe adapted to the hypersurface and write the electromagnetic potential as $A = A_1 e_1 + A_2 e_2 + A_3 e_3 + \phi e_4$ where $e_4=n$; so the potential is a 1-form but not a connection 1-form of any principal bundle over the hypersurface.  One can still look at $F=dA$ decomposed into tangential, mixed and normal components using the Cartan structure equations.  There is geometric naturality to unification of gravity and electromagnetism in this manner and it also provides the gravitational field equations with a simple geometric interpretation.