2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/205554The 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 geometry10 pagesDifferential GeometryGeneral Relativity and Quantum CosmologyHigh Energy Physics - TheoryMathematical Physics53B25 (Primary); 83E15 (Secondary)text