  A complete recipe of measure-preserving diffusions in Euclidean space was
recently derived unifying several MCMC algorithms into a single framework. In
this paper, we develop a geometric theory that improves and generalises this
construction to any manifold. We thereby demonstrate that the completeness
result is a direct consequence of the topology of the underlying manifold and
the geometry induced by the target measure $P$; there is no need to introduce
other structures such as a Riemannian metric, local coordinates, or a reference
measure. Instead, our framework relies on the intrinsic geometry of $P$ and in
particular its canonical derivative, the deRham rotationnel, which allows us to
parametrise the Fokker--Planck currents of measure-preserving diffusions using
potentials. The geometric formalism can easily incorporate constraints and
symmetries, and deliver new important insights, for example, a new complete
recipe of Langevin-like diffusions that are suited to the construction of
samplers. We also analyse the reversibility and dissipative properties of the
diffusions, the associated deterministic flow on the space of measures, and the
geometry of Langevin processes. Our article connects ideas from various
literature and frames the theory of measure-preserving diffusions in its
appropriate mathematical context.
