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 (Σ, g¯\documentclass[12pt]{minimal}
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\begin{document}$$ \overline{g} $$\end{document}). For a flow on the closed Einstein static universe ℝ × S2 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 Kerr- Newman-AdS4 black hole. On the other hand, in the open Einstein static universe ℝ × H2 the situation is more complicated, since the isometry group SL(2, ℝ) of H2 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 class, it has never been studied in the literature before, and represents thus in principle a new black hole solution in AdS4. For a rigidly rotating fluid in ℝ × H2 (holographically dual to the hyperbolic KNAdS4 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 KNAdS4 black hole is conformal to exactly that part of ℝ × H2 in which the fluid velocity does not exceed the speed of light. Thus, the correspondence between AdS gravity and hydrodynamics automatically eliminates the unphysical region. 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 Carter-Plebanski solution to Einstein-Maxwell-AdS gravity. Finally, we show that the alternative description of the hyperbolic KNAdS4 black hole in terms of fluid mechanics on ℝ × S2 or on flat space (both conformal to the open Einstein static universe) is dynamical and consists of a contracting or expanding vortex.
