Minimality Condition and Atmospheric Neutrino Oscillations

A structure is proposed for the mass matrices of the quarks and leptons that arises in a natural way from the assumption that the breaking of $\mathrm{SO}\left(10\right)$ gauge symmetry is achieved by the smallest possible set of vacuum expectation values. This structure explains well many features of the observed spectrum of quarks and leptons. It reproduces the Georgi-Jarlskog mass relations and leads to a charm quark mass in reasonable agreement with data. It also predicts a large mixing angle between ${\nu }_{\mu }$ and ${\nu }_{\tau }$, as suggested by atmospheric neutrino data. The mixing angles of the electron neutrino are predicted to be small.