  In this paper, we give a complete classification of vacuum branes, i.e.,
everywhere umbilical time-like hypersurfaces whose extrinsic curvature is a
constant multiple of the induced metric, K_mn=k g_mn, in D-dimensional static
spacetimes with spatial symmetry G(D-2,K), where G(n,K) is the isometry group
of an n-dimensional space with constant sectional curvature K. D>=4 is assumed.
It is shown that all possible configurations of a brane are invariant under an
isometry subgroup G(D-3,K') for some K'>= K. In particular, configurations of a
brane with non-zero k are always G(D-2,K) invariant, except for those in five
special one-parameter families of spacetimes. Further, such G(D-2,K)-invariant
configurations are allowed only in spacetimes whose Ricci tensors are isotropic
in the two planes orthogonal to each G(D-2,K)-orbit, or for special values of
k, which do not exist in generic cases. On the basis of these results, we prove
the non-existence of a vacuum brane with black hole geometry in static bulk
spacetimes with spatial symmetry G(D-2,K). We also discuss mathematical
implications of these results.
