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

Key Points

• The aim is to explore the relationship between the Navier-Stokes equation and a diffusion process on Riemannian manifolds.
• Investigated the Navier-Stokes equation on Riemannian manifolds with bounded Ricci curvature.
• Applied rolling Brownian motion using a compatible linear connection.
• Computed the time-dependent Ricci curvature related to the Navier-Stokes solutions.
• Established a link between the strain tensor and helicity density through a simple formula in 3D.
• Connected the solutions of the Navier-Stokes equation with the stochastic processes on the manifold.
• Showed that the diffusion process is non-degenerate under the given curvature conditions.

Abstract