  The Fokker-Planck equation is one of the fundamental equations in
nonequilibrium statistical mechanics, and this equation is known to be derived
from the Wasserstein gradient flow equation with a free energy. This gradient
flow equation describes relaxation processes and is formulated on a Riemannian
manifold. Meanwhile contact Hamiltonian systems are also known to describe
relaxation processes. Hence a relation between these two equations is expected
to be clarified, which gives a solid foundation in geometric statistical
mechanics. In this paper a class of contact Hamiltonian systems is derived from
a class of the Fokker-Planck equations on Riemannian manifolds. In the course
of the derivation, the Fokker-Planck equation is shown to be written as a
diffusion equation with a weighted Laplacian without any approximation, which
enables to employ a theory of eigenvalue problems.
