The first attempt at a unified field theory (this refers to unification of gravity and electromagnetism in a unified field) with four spatial dimensions goes back to Gunnar Nordstrom in 1914, a century ago. A great article by Hubert Goenner provides excellent details for this history: HistoryUnifiedFieldTheories

A great deal of effort went into this stream of research but we do not yet have a unified field theory. I suggest that part of the problem of a unified field theory is the lack of restrictions of various kinds that strongly tie the effort to nature. In all my work on an S4 physics I have emphasized two restrictions that I consider empirical and fundamental: that the objective universe has four macroscopic spatial dimensions (this is not consensus view) by an analysis of the observed crystal rotational symmetries of orders 5, 8, 10 and 12; and the compactness of the universe which follows from Gaussian upper bounds on heat kernels for complete noncompact riemannian manifolds. These imply that we should restrict the domain of unified field theories only to four dimensional compact manifolds (five-dimensional spacetime) and in fact that we have concrete evidence that we should in fact seek unified field theories only for a scaled 4-sphere of radius .

These restrictions would presumably make our task easier. Note that the structure of space has gone through extreme attempts at conceptualization. Weyl’s geometry, for example generalized Riemann’s metric geometry. My personal view is that these are curiosities from the point of view of actual description of nature. Weyl’s attempts at unified field theory led to the gauge invariance and non-abelian gauge theories so his ideas obviously have found application in physical description of nature but I think for the geometry of actual spacetime, Riemann’s geometry is appropriate. In fact, the important geometric feature of our actual universe, assuming it compact and four dimensional, is that it be spin in the sense that there be no topological obstructions to lifting the frame bundle to its double cover .

Now if we consider the unified field theory problem on a scaled 4-sphere, a natural choice is to model electromagnetism represented by a 1-form on the full space , and the gravitational metric to be induced from the embedding of a three-dimensional hypersurface . I don’t yet know the right Lagrangian so that the solution is the choice of the physical hypersurface and the Ricci curvature equation for the hypersurface to be exactly identical to the gravitational field equations of Einsten but where the second fundamental form terms are determined by the electromagnetic 1-form. In this picture the issue is not quite a ‘unified field’ where the gravitational potential and electromagnetic potential are tied in a single metric or field (which has been one of the guiding principle of seeking a unification in early 20th century) but rather to begin with a 4-sphere and then reduce gravity to electromagnetism as the choice of a hypersurface. Gravitation arises in this picture not as simply a choice of a metric on a fixed topological space but rather the choice of an embedding and an induced metric for a hypersurface constrained to remain in a 4-sphere the evidence for which we assume are obtained independently on this unification effort.

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