  The purpose of this paper is to establish a probabilistic representation
formula for the Navier--Stokes equations on compact Riemannian manifolds. Such
a formula has been provided by Constantin and Iyer in the flat case of $\mathbb
R^n$ or of $\mathbb T^n$. On a Riemannian manifold, however, there are several
different choices of Laplacian operators acting on vector fields. In this
paper, we shall use the de Rham--Hodge Laplacian operator which seems more
relevant to the probabilistic setting, and adopt Elworthy--Le Jan--Li's idea to
decompose it as a sum of the square of Lie derivatives.
