  We study the spacetime structures of the static solutions in the
$n$-dimensional Einstein-Gauss-Bonnet-$\Lambda$ system systematically. We
assume the Gauss-Bonnet coefficient $\alpha$ is non-negative. The solutions
have the $(n-2)$-dimensional Euclidean sub-manifold, which is the Einstein
manifold with the curvature $k=1,~0$ and -1. We also assume $4{\tilde
\alpha}/\ell^2\leq 1$, where $\ell$ is the curvature radius, in order for the
sourceless solution (M=0) to be defined. The general solutions are classified
into plus and minus branches. The structures of the center, horizons, infinity
and the singular point depend on the parameters $\alpha$, $\ell^2$, $k$, $M$
and branches complicatedly so that a variety of global structures for the
solutions are found. In the plus branch, all the solutions have the same
asymptotic structure at infinity as that in general relativity with a negative
cosmological constant. For the negative mass parameter, a new type of
singularity called the branch singularity appears at non-zero finite radius
$r=r_b>0$. The divergent behavior around the singularity in Gauss-Bonnet
gravity is milder than that around the central singularity in general
relativity. In the $k=1,~0$ cases the plus-branch solutions do not have any
horizon. In the $k=-1$ case, the radius of the horizon is restricted as
$r_h<\sqrt{2\tilde{\alpha}}$ ($r_h>\sqrt{2\tilde{\alpha}}$) in the plus (minus)
branch. There is also the extreme black hole solution with positive mass in
spite of the lack of electromagnetic charge. We briefly discuss the effect of
the Gauss-Bonnet corrections on black hole formation in a collider and the
possibility of the violation of third law of the black hole thermodynamics.
