  We carry on a comprehensive study on static fluid distributions endowed with
hyperbolical symmetry. Their physical properties are analyzed in detail. The
energy density appears to be necessarily negative, which suggests that any
possible application of this kind of fluids requires extreme physical
conditions where quantum effects are expected to play an important role. Also,
it is found that the fluid distribution cannot fill the region close to the
center of symmetry. Such a region may be represented by a vacuum cavity around
the center. A suitable definition of mass function, as well as the Tolman mass
are explicitly calculated. While the former is positive defined, the latter is
negative in most cases, revealing the repulsive nature of gravitational
interaction. A general approach to obtain exact solutions is presented and some
exact analytical solutions are exhibited.
