AI 中文总结
本文更新了1997年关于欧氏空间中taut与Dupin子流形的综述，补充了后续研究成果，详述等参超曲面及两者关系，并给出主要分类结果。
AI 中文摘要
本文是作者于1997年首次发表的论文的更新版本。原论文是对欧几里得空间中 taut 与 Dupin 子流形这两个密切相关领域的综述，而本更新版本包含了自原版发表以来该领域出现的许多研究成果。本文重点在于在恰当的背景下陈述结果并指出未来研究的方向，给出的证明相对较少。文中详细综述了重要的等参超曲面类别，以及 taut 与 Dupin 这两个概念之间的关系。此外，还简要介绍了李球几何中的子流形理论，这是准确陈述许多已知 Dupin 子流形结果所必需的。本文最后详细描述了 Dupin 与 taut 子流形的主要已知分类结果。
英文摘要
This is an updated version of a paper by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. Also included is a brief introduction to submanifold theory in Lie sphere geometry which is needed to state many known results on Dupin submanifolds accurately. The paper concludes with detailed descriptions of the main known classification results for both Dupin and taut submanifolds.
Comments73 pages
