  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.

Analysis of PDEs