  We generalize Yang's theory from the U(1) gauge field to the non-Abelian
$U(1)\times SU(2)_{spin}$ gauge field. Based on this generalization and taking
into account the geometric Pancharatnam phase as well as an effective
Aharonov-Bohm (AB) phase in nonadiabatic noncyclic transport, we calculate the
ensemble average Fourier spectrum of the conductance in disordered mesoscopic
rings connected to two leads. Our theory can explain the experimental results
reported by Morpurgo {\sl et al.} [Phys. Rev. Lett. {\bf 80}, 1050 (1998)]
satisfactorily. In particular, we clarify that the experimentally observed
splitting, as well as some structure on the sides of the main peak in the
ensemble average Fourier spectrum, stem from the nonadiabatic noncyclic
Pancharatnam phase and the effective AB phase, both being dependent on
spin-orbit coupling.
