Normal variation of a hypersurface of with can be described by a function by applying the exponential map along the normal unit vector , i.e. . We are interested in considering deformations of to . Let be tangential coframes and it is known that their images under this deformations are Jacobi vector fields:
where are parallel translates of along the normal geodesic where
Now the gravitational lagrangian for hypersurfaces of is the integral of the scalar curvature which is known explicitly in terms of the second fundamental form
where is the mean curvature and is the squared norm of the second fundamental form. In analogy with the derivation of the Einstein field equations, we consider the normal variations by arbitrary functions and then consider the critical points of the function under the variations. This I will claim will produce the correct graviational field equations because we already know that the universe is a compact scaled four-sphere.
The implications of this picture are obviously extensive — for this points to four dimensional matter that is completely not accounted for by the established three dimensional theories of quantum mechanics and general relativity. Quantum mechanics is likely a linear approximation of the correct physics.