We investigate the topology of Schwarzschilds black hole through the immersion of this space-time in spaces of higher dimension. Through the immersions of Kasner and Fronsdal we calculate the extension of the Schwarzschilds black hole.
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 Schwarzschilds 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.
The braneworlds models were inspired partly by Kaluza-Kleins theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modif
ications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newtons law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature,(for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrums braneworld model.
