  Properties of $n(\ge 5)$-dimensional static wormhole solutions are
investigated in Einstein-Gauss-Bonnet gravity with or without a cosmological
constant $\Lambda$. We assume that the spacetime has symmetries corresponding
to the isometries of an $(n-2)$-dimensional maximally symmetric space with the
sectional curvature $k=\pm 1, 0$. It is also assumed that the metric is at
least $C^{2}$ and the $(n-2)$-dimensional maximally symmetric subspace is
compact. Depending on the existence or absence of the general relativistic
limit $\alpha \to 0$, solutions are classified into general relativistic (GR)
and non-GR branches, respectively, where $\alpha$ is the Gauss-Bonnet coupling
constant. We show that a wormhole throat respecting the dominant energy
condition coincides with a branch surface in the GR branch, otherwise the null
energy condition is violated there. In the non-GR branch, it is shown that
there is no wormhole solution for $k\alpha \ge 0$. For the matter field with
zero tangential pressure, it is also shown in the non-GR branch with
$k\alpha<0$ and $\Lambda \le 0$ that the dominant energy condition holds at the
wormhole throat if the radius of the throat satisfies some inequality. In the
vacuum case, a fine-tuning of the coupling constants is shown to be necessary
and the radius of a wormhole throat is fixed. Explicit wormhole solutions
respecting the energy conditions in the whole spacetime are obtained in the
vacuum and dust cases with $k=-1$ and $\alpha>0$.
