四维超曲面的高维嵌入：余维1嵌入的时间发展
Higher-dimensional Embedding of Four-dimensional Hypersurfaces: Chronological Development of Co-dimension-1 Embedding
摘要：
在这里，我们提出了一个概述的工作，多年来所做的嵌入超曲面在一个更高的维度空间，特别是四维超曲面嵌入在五维时空。嵌入的概念是由几何学家从高斯开始发展的。在物理学中的应用开始于史瓦西度规的嵌入，并通过卡鲁扎和克莱因的作品。在五维时空中嵌入四维超曲面的计划随着韦森及其合作者的诱导物质理论的形成而增强。紧随其后的是关于膜世界场景的研究。与此同时，四维超曲面嵌入五维时空的数学方面也作为坎贝尔-马加德定理的一个应用得到了发展。在本文中，我们试图有一个全景的主要工作在这些领域所做的，并试图集中在这方面的最新趋势。
英文摘要：
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.
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