Generic conformally flat hypersurfaces and surfaces in 3-sphere  

作　　者：Yoshihiko Suyama 

机构地区：[1]Department of Applied Mathematics,Faculty of Sciences,Fukuoka University,Fukuoka 814-0180,Japan

出　　处：《Science China Mathematics》2020年第12期2439-2474,共36页中国科学：数学（英文版）

摘　　要：The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere.The conformal structure of generic conformally flat(local-)hypersurfaces is characterized as conformally flat(local-)3-metrics with the Guichard condition.Then,there is a certain class of orthogonal analytic(local-)Riemannian 2-metrics with constant Gauss curvature-1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition.In this paper,we firstly relate 2-metrics of the class to surfaces in the 3-sphere:for a 2-metric of the class,a 5-dimensional set of(non-isometric)analytic surfaces in the 3-sphere is determined such that any surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surface and the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean4-space.Secondly,we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces.

关 键 词：conformally flat hypersurface system of evolution equations Guichard net integrability condition surface in 3-sphere 

分 类 号：O186.11[理学—数学]