  We consider a thick p-brane embedded in an n-dimensional spacetime possessing
radial symmetry in the directions orthogonal to the brane. We first consider a
static brane, and find a general fine tuning relationship between the brane and
bulk parameters required for the brane to be flat. We then consider the
cosmology of a time dependent brane in a static bulk, and find the Friedmann
equation for the brane scale factor a(t). The singularities that would
ordinarily arise when considering arbitrary codimensions are avoided by
regularizing the brane, giving it a finite profile in the transverse
dimensions. However, since we consider the brane to be a strictly local defect,
we find that the transverse dimensions must have infinite volume, and hence
gravity cannot be localized on the brane without resorting to some infra-red
cutoff.
