  In the note added in proof of the seminal paper [Groups of diffeomorphisms
andthe motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163],
Ebinand Marsden introduced the so-called correct Laplacian for the
Navier-Stokes equationon a compact Riemannian manifold. In the spirit of
Brenier's generalized flows forthe Euler equation, we introduce a class of
semimartingales on a compact Riemannianmanifold. We prove that these
semimartingales are critical points to the correspondingkinetic energy if and
only if its drift term solves weakly the Navier-Stokes equationdefined with
Ebin-Marsden's Laplacian. We also show that for the torus case,classical
solutions of the Navier-Stokes equation realize the minimum of the
kineticenergy in a suitable class.
