  Propagation of fermions in spacetime requires a spin connection, which can be
split into a universal gravitational part and a non-universal ``contorsion''
part. The latter is non-dynamical and can be eliminated from the theory,
leaving an effective four-fermion interaction with unknown coupling constants.
The generic form of the contorsion-fermion coupling, and thus the four-fermion
interaction, breaks chiral symmetry. This interaction affects all fermions --
in particular neutrinos passing through matter will notice a contribution to
their effective Hamiltonian, just like the MSW effect coming from weak
interactions, but diagonal in the mass basis rather than the flavor basis. Then
there is a possibility that this geometrical contribution is not negligible for
neutrinos passing through normal matter, provided the coupling constants are
not too small. We calculate the matter potential due to this interaction and
thus write the conversion and survival probabilities including its effect. We
plot conversion probabilities of $\nu_\mu$ to $\nu_e$ and $\nu_\tau$ for a
baseline of 1300 km, as well as their dependence on the CP phase, with and
without the geometrical interaction. We also plot the survival probability of
reactor $\bar{\nu}_e$ for different baselines.
