  We study conformal properties of the quantum kinetic equations in curved
spacetime. A transformation law for the covariant Wigner function under
conformal transformations of a spacetime is derived by using the formalism of
tangent bundles. The conformal invariance of the quantum corrected Vlasov
equation is proven. This provides a basis for generating new solutions of the
quantum kinetic equations in the presence of gravitational and other external
fields. We use our method to find explicit quantum corrections to the class of
locally isotropic distributions, to which equilibrium distributions belong. We
show that the quantum corrected stress--energy tensor for such distributions
has, in general, a non--equilibrium structure. Local thermal equilibrium is
possible in quantum systems only if an underlying spacetime is conformally
static (not stationary). Possible applications of our results are discussed.
