  We show that the classification of shearless and incompressible stationary
fluid flows on ultrastatic manifolds is equivalent to classifying the
isometries of the spatial sections. For a flow on R x S$^2$ this leaves only
one possibility, since on the 2-sphere all Killing fields are conjugate to each
other, and it is well-known that the gravity dual of such a (conformal) fluid
is the spherical KNAdS$_4$ black hole. On the other hand, in R x H$^2$ the
situation is more complicated, since the isometry group of H$^2$ admits
elliptic, parabolic and hyperbolic elements. One might thus ask what the
gravity duals of the flows corresponding to these three different cases are.
Answering this question is one of the scopes of this paper. In particular we
identify the black hole dual to a fluid that is purely translating on the
hyperbolic plane. Although this lies within the Carter-Plebanski (CP) class, it
has never been studied in the literature before, and represents thus in
principle a new black hole solution in AdS$_4$. For a rigidly rotating fluid in
R x H$^2$ (holographically dual to the hyperbolic KNAdS$_4$ solution), there is
a certain radius where the velocity reaches the speed of light, and thus the
fluid can cover only the region within this radius. Quite remarkably, it turns
out that the boundary of the hyperbolic KNAdS$_4$ black hole is conformal to
exactly that part of R x H$^2$ in which the fluid velocity does not exceed the
speed of light. We extend these results to establish a precise mapping between
possible flows on ultrastatic spacetimes (with constant curvature spatial
sections) and the parameter space of the CP solution. Finally, we show that the
alternative description of the hyperbolic KNAdS$_4$ black hole in terms of
fluid mechanics on R x S$^2$ or on flat space is dynamical and consists of a
contracting or expanding vortex.
