  We consider the motion of an incompressible viscous fluid on a compact
Riemannian manifold $\sM$ with boundary. The motion on $\sM$ is modeled by the
incompressible Navier-Stokes equations, and the fluid is subject to pure or
partial slip boundary conditions of Navier type on $\partial\sM$. We establish
existence and uniqueness of strong as well as weak (variational) solutions for
initial data in critical spaces. Moreover, we show that the set of equilibria
consists of Killing vector fields on $\sM$ that satisfy corresponding boundary
conditions, and we prove that all equilibria are (locally) stable. In case
$\sM$ is two-dimensional we show that solutions with divergence free initial
condition in $L_2(\sM; T\sM)$ exist globally and converge to an equilibrium
exponentially fast.
