The possibility of a description of the fundamental interactions of physics, including gravitation, based upon the assumption of 6 real extra dimensions is presented. The usual 4-dimension space-time, a curved surface with the Lorentz group as local symmetry, is embedded in a larger flat 10-dimension space. Through a fundamental assumption about the geometry of the orthogonal 6-d space in every point of the 4-d surface, there are two possibilities for classifying the physical states, corresponding to two types of particles: 1) hadrons, experiencing a gauge field associated to a real symmetry group GH(6), isomorphous to SU(3), which is identified with the strong interaction, and 2) leptons experiencing another gauge field associated with a real symmetry group GL(6), isomorphous to SU(2) × U(1) but different from the usual electroweak coupling. In addition, both hadrons and leptons are subject to weak and electromagnetic interactions plus a scalar BEH-like coupling, with the respective real symmetries SO(3), SO(2), SO(1), isomorphous to SU(2), U(1), I(1). This description can be extended so as to include gravitation; postulating a minimal Lagrangian in the full 10-d space, the equations of motion are derived. They imply the existence of a set of additional vector-type fields which do not act the same way upon hadrons and leptons, thus inducing a violation of the equivalence principle.