Living inside a hedgehog: higher-dimensional solutions that localize gravity

We consider spherically symmetric higher-dimensional solutions of Einstein' s equations with a bulk cosmological constant and n transverse dimensions. In contrast to the case of one or two extra dimensions we find no solutions that localize gravity when n greater than or equal to 3, for strictly loca l topological defects. We discuss global topological defects that lead to t he localization of gravity and estimate the corrections to Newton's law. We show that the introduction of a bulk "hedgehog" magnetic field leads to a regular geometry and localizes gravity on the 3-brane with either a positiv e, zero or negative bulk cosmological constant. The corrections to Newton's law on the 3-brane are parametrically the same as for the case of one tran sverse dimension. (C) 2000 Elsevier Science B.V. All rights reserved.