It is known that an object translating parallel to a soft wall in a viscous fluid produces
hydrodynamic stresses that deform the wall, which in turn results in a lift force on the
object. Recent experiments with cylinders sliding under gravity near a soft incline, which
confirmed theoretical arguments for the lift force, also reported an unexplained steady-state
rotation of the cylinders [B. Saintyves et al., Proc. Natl. Acad. Sci. USA 113, 5847 (2016)].
Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder
that translates in a viscous fluid parallel to a soft wall at constant speed and separation
distance must also rotate in order to remain free of torque. Using the Lorentz reciprocal
theorem, we show analytically that for small deformations of the elastic layer, the angular
velocity of the cylinder scales with the cube of the sliding speed. These predictions are
confirmed numerically. We then apply the theory to the gravity-driven motion of a cylinder
near a soft incline and find qualitative agreement with the experimental observa