One can extract from Jonah Miller’s Kaluza-Klein paper (jonah-miller-kaluza-klein-theory) the following result:
If the metric for a four (space) dimensional space can be written as:
then the components of the curvature 2-form are:
and other terms and the scalar curvature in the snapshot:
Now we can consider this a generic theorem for four space-dimensional manifolds and specialize to four-sphere hypersurfaces with the following special : we take the second fundamental form 3×3 matrix and diagonalize it, setting . Miller’s arguments for unification of electromagnetism and gravitation is unchanged given that the vector potential is given in this special form.