  In this work we show that (real space) curvature in the geometry of curved
waveguides with Rashba spin orbit interaction (RSOI) can lead to out of plane
spin accumulations. We first derive the RSOI Hamiltonian on arbitrarily curved
surfaces. We then analyze the effects of curvature with two distinct methods.
We first apply an adiabatic approximation on gently curved, planar waveguides
lying flat on the $xy$ plane to show that analogous to the acceleration of the
charge carriers by an electric field, the change in momentum direction of the
charge carriers as they move along the waveguide leads to an out of plane spin
accumulation. We then use the Heisenberg equations of motion to establish the
relationships between spin currents and accumulations on non-planar waveguides.
These relations predict the existence of out of plane spin accumulation on
asymmetrically curved, non-planar waveguides. We finally solve for the
eigenstates on such waveguides numerically, and present numerical results to
verify our earlier analytic predictions.
