  We consider the motion of an incompressible viscous fluid that completely
covers a smooth, compact and embedded hypersurface $\Sigma$ without boundary
and flows along $\Sigma$. Local-in-time well-posedness is established in the
framework of $L_p$-$L_q$-maximal regularity. We characterize the set of
equilibria as the set of all Killing vector fields on $\Sigma$ and we show that
each equilibrium on $\Sigma$ is stable. Moreover, it is shown that any solution
starting close to an equilibrium exists globally and converges at an
exponential rate to a (possibly different) equilibrium as time tends to
infinity.
