Features of Quark and Lepton Mixing from Differential Geometry of Curves on Surfaces

It is noted that the CKM matrix elements for both quarks and leptons as conceived in the Dualized Standard Model (DSM) can be interpreted as direction cosines obtained by moving the Darboux trihedron (a 3-frame) along a trajectory on a sphere traced out through changing energy scales by a 3-vector factorized from the mass matrix. From the `Darboux’ analogues of the well-known Serret–Frenet formulae for space curves, it is seen that the corner elements ($V_{ub},V_{td}$ for quarks, and $U_{e3},U_{\tau 1}$ for leptons) are associated with the (geodesic) torsion, while the other off-diagonal elements ($V_{us},V_{cd}$ and $V_{cb},V_{ts}$ for quarks, and $U_{e2},U_{\mu 1}$ and $U_{\mu 3},U_{\tau 2}$ for leptons) with the (respectively geodesic and normal) curvatures of the trajectory. From this it follows that (i) the corner elements in both matrices are much smaller than the other elements, (ii) the $U_{\mu 3},U_{\tau 2}$ elements for the lepton CKM matrix are much larger than their counterparts in the quark matrix. Both these conclusions are strongly borne out by experiment, for quarks in hadron decays and for leptons in neutrino oscillations, and by previous explicit calculations within the DSM scheme.