Topological and Geometrical Properties of Brane-Worlds

TL;DR

This paper investigates how the geometry and topology of a brane-world influence the properties of the surrounding bulk space, revealing topological differences and signature changes through immersion formalism.

Contribution

It introduces a method to analyze topological and geometrical differences in brane-worlds immersed in higher-dimensional spaces, focusing on Schwarzschild and Kruskal extensions.

Findings

Existence of topological differences in immersed brane-worlds.

Signature change of the bulk with different topologies.

Application to Schwarzschild and Kruskal brane-worlds.

Abstract

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.