  We derive the chiral kinetic theory under the presence of a gravitational
Riemann curvature. It is well-known that in the chiral kinetic theory there
inevitably appears a redundant ambiguous vector corresponding to the choice of
the Lorentz frame. We reveal that on top of this conventional frame choosing
vector, higher-order quantum correction to the chiral kinetic theory brings an
additional degrees of freedom 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, for which inhomogeneous vorticity or
temperature are effectively represented by spacetime metric tensor. 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.
