  In order to find out whether empty singular boundaries can arise in higher
dimensional Gravity, we study the solution of Einstein's equations consisting
in a ($N+2$)-dimensional static and hyperplane symmetric perfect fluid
satisfying the equation of state $\rho=\eta\, p$, being $\rho$ an arbitrary
constant and $N\geq2$. We show that this spacetime has some weird properties.
In particular, in the case $\eta>-1$, it has an empty (without matter)
repulsive singular boundary.
  We also study the behavior of geodesics and the Cauchy problem for the
propagation of massless scalar field in this spacetime. For $\eta>1$, we find
that only vertical null geodesics touch the boundary and bounce, and all of
them start and finish at $z=\infty$; whereas non-vertical null as well as all
time-like ones are bounded between two planes determined by initial conditions.
We obtain that the Cauchy problem for the propagation of a massless scalar
field is well-posed and waves are completely reflected at the singularity, if
we only demand the waves to have finite energy, although no boundary condition
is required.
