Higher-dimensional Embedding of Four-dimensional Hypersurfaces: Chronological Development of Co-dimension-1 Embedding

TL;DR

This paper reviews the historical and recent developments in embedding four-dimensional hypersurfaces into five-dimensional spaces, highlighting mathematical foundations and physical applications like braneworlds and induced matter theory.

Contribution

It provides a comprehensive overview of the evolution and recent trends in higher-dimensional hypersurface embedding, integrating mathematical theorems and physical models.

Findings

Historical development from Gauss to modern theories

Connection between embedding and physical models like braneworlds

Recent focus on mathematical theorems such as Campbell-Magaard

Abstract

Here we present an overview of the work done over the years on the embedding of hypersurfaces in a space of higher dimension, with particular reference to the embedding of four-dimensional hypersurfaces in five-dimensional space-times. The concept of embedding was developed by geometers starting from Gauss. Applications in physics started with the embedding of the Schwarzschild metric and through the works of Kaluza and Klein. The program on the embedding of four-dimensional hypersurfaces in five-dimensional spacetimes gathered strength with the formulation of the induced matter theory by Wesson and his collaborators. Close on its heels followed the works on the braneworld scenario. Simultaneously, the mathematical aspects of the embedding of four-dimensional hypersurfaces in five-dimensional spacetimes were developed as an application of the Campbell-Magaard theorem. In this paper we have tried to have a panoramic view of the major works done in these fields and have tried to focus on the recent trends in this context.