  We derive the chiral kinetic theory under the presence of a gravitational
Riemann curvature. We reveal that on top of the conventional frame choosing
vector, higher order quantum correction to the chiral kinetic theory brings an
additional ambiguity to specify the distribution function. Based on this
framework, we derive new types of fermionic transport, that is, the charge
current and energy-momentum tensor induced by the gravitational Riemann
curvature. Such novel phenomena arise not only under genuine gravity but also
in a (pseudo-)relativistic fluid with inhomogeneous vorticity or temperature.
It is especially found that the charge and energy currents are antiparallelly
induced by an inhomogeneous fluid vorticity (more generally, by the Ricci
tensor ${R_0}^i$), as a consequence of the spin-curvature coupling. We also
briefly discuss possible applications to Weyl/Dirac semimetals and heavy-ion
collision experiments.
