The Navier-Stokes equations can be written on a Riemannian manifold as:
$$\dot{u}+\nabla\_u u+ \Delta u=(df)^\* $$
$$d^\* u=0$$
where $\nabla$ is the Levi-Civita connection, $u$ is a vector field, $\Delta$ is the Laplacian, $df$ is the differential of $f$, $(df)^\* $ is the dual of $df$ via the metric, and $d^\*u$ is the divergence of $u$.


The problem is due to Antoine Balan.


Do you have references?



Navier-Stokes equations in Riemannian geometry
The answer and comments about Arnold and Marsden papers are a little off side. They concern the equation of inviscid fluids, called Euler equation. This differs from Navier-Stokes by the highest-order derivatives $\Delta u$. This changes completely the functional analysis background. Also, Euler equation has a geometrical interpretation (geodesics on the group of measure-preserving diffeomorphisms), whereas Navier-Stokes has not.


I am not aware of references for Navier-Stokes on manifolds. However, I don't think that this is a real problem. What has been important so far for Navier-Stokes is the space dimension and the embedding theorems we have between functional spaces like Sobolev, Besov and others. For instance, the Cauchy problem must be globally well-posed on every compact surface, and locally well-posed on $3$-manifolds.


