Ikeda-Watanabe’s Connection, Brownian Motion and Navier-Stokes Equation on Riemannian Manifolds

We consider the Navier-Stokes equation on a Riemannian manifold with the Ricci curvature bounded below. In stochastic analysis, a non-degenerate diffusion process on a Riemannian manifold was obtained by rolling Brownian motion with respect to a suitable metric compatible linear connection, which was introduced by N. Ikeda and S. Watanabe about 40 years ago. To each solution of the Navier-Stokes equation, we associate such a connection and compute the related time-dependent Ricci curvature, which allow us to obtain a link with the strain tensor and the helicity density in a simple formula in the case of dimension 3.