  We study the geometrical and topological properties of the bulk (environment
space) when we modify the geometry or topology of a brane-world. Through the
characterization of a spherically symmetric space-time as a local brane-world
immersed into six dimensional pseudo-Euclidean spaces, with different
signatures of the bulk, we investigate the existence of a topological
difference in the immersed brane-world. In particular the Schwarzschild's
brane-world and its Kruskal (or Fronsdal) brane-world extension are examined
from point of view of the immersion formalism. We prove that there is a change
of signature of the bulk when we consider a local isometric immersion and
different topologies of a brane-world in that bulk.
