Geometric Unification of the Fundamental Interactions

Kaluza-Klein theory is modified, by changing the usual product topology of the total space by the embedding space of the space-time. The Einstein-Hilbert action applied to the higher dimensional space is maintained, but the metric is derived from the embedding. The internal space is not compact as it is generated by the extra dimensions of the embedding space and the isometry group plays the role of the gauge symmetry. It follows from the embedding assumption that the gauge potentials are of geometrical origin, given by the third fundamental form of the embedded space-time. The Lagrangian of the total metric decomposes in the gravitational plus the Yang-Mills plus an interaction term determined by the second fundamental form (or extrinsic curvature). The advantage over the original Kaluza-Klein is that the problem with fermions chirality at the electroweak scale do nor appear and the hierarchy of the fundamental interactions is resolved