The purpose of this note is to establish a probabilistic representation formula for Navier–Stokes equations on a compact Riemannian manifold. To this end, we first give a geometric interpretation of Constantin and Iyer's representation formula for the Navier–Stokes equation, then extend it to a compact Riemannian manifold. We shall use Elworthy–Le Jan–Li's idea to decompose de Rham–Hodge Laplacian operator on a manifold as a sum of the square of vector fields. MSC 2010: 35Q30, 58J65 Keywords: Navier–Stokes equations, stochastic representation, de Rham–Hodge Lapla-cian, stochastic flow, pull-back vector field