  We analytically calculate the spin-dependent electronic conductance through a
one-dimensional ballistic ring in the presence of an inhomogeneous magnetic
field and identify signatures of geometric and Berry phases in the general
non-adiabatic situation. For an in-plane magnetic field, we rigorously prove
the spin-flip effect presented in Frustaglia et al., Phys. Rev. Lett. 87,
256602 (2001), which allows to control and switch the polarization of outgoing
electrons by means of an Aharonov-Bohm flux, and derive analytical expressions
for the energy-averaged magneto-conductance. Our results support numerical
calculations for two-dimensional ballistic rings presented in the second paper
(Frustaglia et al., submitted to Phys. Rev. B) of this series.
