  This article discusses the relativistic kinetic theory for a simple
collisionless gas from a geometric perspective. We start by reviewing the rich
geometrical structure of the tangent bundle TM of a given spacetime manifold,
including the splitting of the tangent spaces of TM into horizontal and
vertical subspaces and the natural metric and symplectic structure it induces
on TM. Based on these structures we introduce the Liouville vector field L and
a suitable Hamiltonian function H on TM. The Liouville vector field turns out
to be the Hamiltonian vector field associated to H. On the other hand, H also
defines the mass shells as Lorentzian submanifolds of the tangent bundle. A
simple collisionless gas is described by a distribution function on a
particular mass shell, satisfying the Liouville equation. Together with the
Liouville vector field the distribution function can be thought of as defining
a fictitious incompressible fluid on the mass shells, with associated conserved
current density. Flux integrals of this current density provide the averaged
properties of the gas, while suitable fibre integrals of the distribution
function define divergence-free tensor fields on the spacetime manifold such as
the current density and stress-energy tensor. Finally, we discuss the
relationship between symmetries of the spacetime manifold and symmetries of the
distribution function.
  As a first application of our formalism we derive the most general
spherically symmetric distribution function on any spherically symmetric
spacetime and write the Einstein-Liouville equations as effective field
equations on the two-dimensional radial manifold. As a second application we
derive the most general collisionless distribution function on a Kerr black
hole spacetime background.
