  We shall prove dispersive and smoothing estimates for Bochner type laplacians
on some non-compact Riemannian manifolds with negative Ricci curvature, in
particular on hyperbolic spaces. These estimates will be used to prove
Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We
shall also discuss the uniqueness of Leray weak solutions in the two
dimensional case.
