# Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

## TL;DR

This paper investigates suitable Laplace operators for the Navier-Stokes equations on Riemannian manifolds using Nash embedding, comparing de Rham-Hodge and Ebin-Marsden Laplacians, and provides a probabilistic representation formula.

## Contribution

It elucidates the roles of different Laplace operators in Navier-Stokes equations on manifolds and derives a probabilistic formula involving the de Rham-Hodge Laplacian.

## Key findings

- Comparison of Laplace operators on vector fields for Navier-Stokes
- Probabilistic representation formula for Navier-Stokes on manifolds
- Insights into the geometric structure of fluid dynamics equations

## Abstract

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC 2010: 35Q30, 58J65