<p>Fermions act as sources of spacetime torsion. However, this torsion is non-dynamical and can be eliminated using its equations of motion. The resulting theory features an effective four-fermion interaction in a torsion-free background. Generically this interaction is non-universal and violates parity. When neutrinos propagate through matter, they experience the effect of this geometrical interaction, which is similar to the MSW effect, but diagonal in the mass basis. Since this quartic interaction term varies linearly with matter density, its effect will be more prominent for atmospheric neutrinos specially for upward going atmospheric neutrinos. We investigate the effect of spacetime geometry on <i>ν</i>_<i>μ</i> → <i>ν</i>_<i>τ</i> and <i>ν</i>_<i>μ</i> → <i>ν</i>_<i>e</i> conversion probability and <i>ν</i>_<i>μ</i> survival probability, by solving the Schrödinger equation using the Preliminary Reference Earth Model (PREM) density profile of the Earth. We also study the dependence of atmospheric neutrino oscillation probability on the CP phase angle in presence of this effect. We further provide a discussion of the MSW and parametric resonances in presence of the geometrical interaction.</p>