  An exhaustive classification of certain class of static solutions for the
five-dimensional Einstein-Gauss-Bonnet theory in vacuum is presented. The class
of metrics under consideration is such that the spacelike section is a warped
product of the real line with a nontrivial base manifold. It is shown that for
generic values of the coupling constants the base manifold must be necessarily
of constant curvature, and the solution reduces to the topological extension of
the Boulware-Deser metric. It is also shown that the base manifold admits a
wider class of geometries for the special case when the Gauss-Bonnet coupling
is properly tuned in terms of the cosmological and Newton constants. This
freedom in the metric at the boundary, which determines the base manifold,
allows the existence of three main branches of geometries in the bulk. For
negative cosmological constant, if the boundary metric is such that the base
manifold is arbitrary, but fixed, the solution describes black holes whose
horizon geometry inherits the metric of the base manifold. If the base manifold
possesses a negative constant Ricci scalar, two different kinds of wormholes in
vacuum are obtained. For base manifolds with vanishing Ricci scalar, a
different class of solutions appears resembling "spacetime horns". There is
also a special case for which, if the base manifold is of constant curvature,
due to certain class of degeneration of the field equations, the metric admits
an arbitrary redshift function. For wormholes and spacetime horns, there are
regions for which the gravitational and centrifugal forces point towards the
same direction. All these solutions have finite Euclidean action, which reduces
to the free energy in the case of black holes, and vanishes in the other cases.
Their mass is also obtained from a surface integral.
