  A class of linear kinetic Fokker-Planck equations with a non-trivial
diffusion matrix and with periodic boundary conditions in the spatial variable
is considered. After formulating the problem in a geometric setting, the
question of the rate of convergence to equilibrium is studied within the
formalism of differential calculus on Riemannian manifolds. Under explicit
geometric assumptions on the velocity field, the energy function and the
diffusion matrix, it is shown that global regular solutions converge in time to
equilibrium with exponential rate. The result is proved by estimating the time
derivative of a modified entropy functional, as recently proposed by Villani.
For spatially homogeneous solutions the assumptions of the main theorem reduce
to the curvature bound condition for the validity of logarithmic Sobolev
inequalities discovered by Bakry and Emery. The result applies to the
relativistic Fokker-Planck equation in the low temperature regime, for which
exponential trend to equilibrium was previously unknown.
