The formulation of the Navier-Stokes equations on Riemannian manifolds

TL;DR

This paper investigates how to properly formulate the Navier-Stokes equations on Riemannian manifolds, comparing different approaches and advocating for a specific formulation as the most appropriate generalization.

Contribution

The paper analyzes various formulations of Navier-Stokes on manifolds and argues for adopting the Ebin and Marsden formulation as the correct generalization.

Findings

Several formulations of Navier-Stokes on manifolds are compared.

Arguments favor the Ebin and Marsden formulation as the most suitable.

The paper clarifies the mathematical foundations for fluid dynamics on curved spaces.

Abstract

We consider the generalization of the Navier-Stokes equations from $\mathbb R^n$ to the Riemannian manifolds. There are inequivalent formulations of the Navier-Stokes equations on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds.