  Exact vacuum solutions with a nontrivial torsion for the
Einstein-Gauss-Bonnet theory in five dimensions are constructed. We consider a
class of static metrics whose spacelike section is a warped product of the real
line with a nontrivial base manifold endowed with a fully antisymmetric
torsion. It is shown requiring solutions of this sort to exist, fixes the
Gauss-Bonnet coupling such that the Lagrangian can be written as a Chern-Simons
form. The metric describes black holes with an arbitrary, but fixed, base
manifold. It is shown that requiring its ground state to possess unbroken
supersymmetries, fixes the base manifold to be locally a parallelized
three-sphere. The ground state turns out to be half-BPS, which could not be
achieved in the absence of torsion in vacuum. The Killing spinors are
explicitly found.
