We consider a certain
invariantly defined nonlinear system of partial differential equations on a Riemannian
manifold. Since a special case describes a steady, irrotional, compressible flow on the
manifold, it is natural to refer to the (square of) the pointwise norm of the solution
as the speed of the flow and to the density of the flow. Under appropriate restrictions
on the density, the system is elliptic and we obtain a sub-elliptic estimate and a
maximum principle for the speed of the flow in terms of the curvature of the
manifold.