  A consistent description of the fundamental interactions of particle physics
based upon the assumption of 6 real extra dimensions is presented. The usual
4-dimension space-time, a curved hypersurface with the Lorentz group as local
symmetry, is assumed to be embedded in a larger flat 10-dimension space-time.
Two additional assumptions are made: (i) the orthogonal 6-d space in any point
of the hypersurface is invariant under the orthogonal group SO(6) or one of its
subgroups, (ii) in that space only the invariant, or distinguished, subgroups
of SO(6) can be symmetry groups for the physical states. There are only two
such possibilities, each of them corresponding to one type of particles: (i)
hadrons, experiencing a gauge field associated to a real symmetry group GH(6) ,
isomorphous to SU(3) and identified with the strong interaction, and (ii)
leptons experiencing another gauge field associated with a real symmetry group
GL(6), isomorphous to SU(2)xU(1) but different from the usual electro-weak
coupling. Moreover both hadrons and leptons are subject to weak and
electromagnetic interactions plus a scalar Higgs-type coupling. That
description can be extended so as to include gravitation. Postulating a minimal
Lagrangian in the full 10-dimension space-time leads to introduce one effective
additional vector-type field, which does not act the same way upon hadrons and
leptons, thus provoking an apparent violation of the equivalence principle.
