  Traversable wormhole are primarily useful as "gedanken-experiments" and as a
theoretician's probe of the foundations of general relativity. In this work, we
analyse the possibility of having tunnels in a hyperbolic spacetime. We obtain
exact solutions of static and pseudo-spherically symmetric spacetime tunnels by
adding exotic matter to a vacuum solution referred to as a degenerate solution
of class A. The physical properties and characteristics of these intriguing
solutions are explored, and through the mathematics of embedding it is shown
that particular constraints are placed on the shape function, that differ
significantly from the Morris-Thorne wormhole. In particular, it is shown that
the energy density is always negative and the radial pressure is positive, at
the throat, contrary to the Morris-Thorne counterpart. Specific solutions are
also presented by considering several equations of state, and by imposing
restricted choices for the shape function or the redshift function.
