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

In this paper, we give a complete classification of vacuum branes, i.e.,everywhere umbilical time-like hypersurfaces whose extrinsic curvature is aconstant multiple of the induced metric, K_mn=k g_mn, in D-dimensional staticspacetimes with spatial symmetry G(D-2,K), where G(n,K) is the isometry groupof 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 anisometry subgroup G(D-3,K') for some K'>= K. In particular, configurations of abrane with non-zero k are always G(D-2,K) invariant, except for those in fivespecial one-parameter families of spacetimes. Further, such G(D-2,K)-invariantconfigurations are allowed only in spacetimes whose Ricci tensors are isotropicin the two planes orthogonal to each G(D-2,K)-orbit, or for special values ofk, which do not exist in generic cases. On the basis of these results, we provethe non-existence of a vacuum brane with black hole geometry in static bulkspacetimes with spatial symmetry G(D-2,K). We also discuss mathematicalimplications of these results.Comment: 42 pages in LaTeX with the PTP style. No figure. Minor errors are corrected, and notations are partially change