Vacuum Branes in D - Dimensional Static Spacetimes with Spatial Symmetry IO ( D − 2 ) , O ( D − 1 ) or O + ( D −

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, Kμν = σgμν , 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 σ 6= 0 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 σ, 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.