Mathematical Support to Braneworld Theory

TL;DR

This paper provides a mathematical proof supporting the consistency of braneworld theory by establishing integrability conditions for space-time submanifolds in high-dimensional pseudo-Euclidean spaces.

Contribution

It offers a new proof of the fundamental theorem of submanifolds tailored for semi-Riemannian manifolds, underpinning the mathematical foundation of braneworld models.

Findings

Established integrability conditions for space-time submanifolds

Provided a new proof of the fundamental theorem of submanifolds

Supported the mathematical consistency of braneworld theory

Abstract

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold in a space of high dimension, the bulk, solution of Einstein's equations in high dimension. In this paper we approach the mathematical consistency of this theory with a new proof of the fundamental theorem of submanifolds for case of semi-Riemannian manifolds. This theorem consist an essential mathematical support for this new theory. We find the integrability conditions for the existence of space-time submanifolds in a pseudo-Euclidean space. Keywords: Submanifolds, Braneworld, Pseudo-Riemannian geometry