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Here we present an overview of the work done over the years on the embeddingof hypersurfaces in a space of higher dimension, with particular reference tothe embedding of four-dimensional hypersurfaces in five-dimensionalspace-times. The concept of embedding was developed by geometers starting fromGauss. Applications in physics started with the embedding of the Schwarzschildmetric and through the works of Kaluza and Klein. The program on the embeddingof four-dimensional hypersurfaces in five-dimensional spacetimes gatheredstrength with the formulation of the induced matter theory by Wesson and hiscollaborators. Close on its heels followed the works on the braneworldscenario. Simultaneously, the mathematical aspects of the embedding offour-dimensional hypersurfaces in five-dimensional spacetimes were developed asan application of the Campbell-Magaard theorem. In this paper we have tried tohave a panoramic view of the major works done in these fields and have tried tofocus on the recent trends in this context.

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