Generic conformally flat hypersurfaces and surfaces in 3-sphere
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https://doi.org/10.1007/s11425-020-1761-9
Generic conformally flat hypersurfaces and surfaces in 3-sphere Dedicated to Professor Qing-Ming Cheng for His 60th Birthday
Yoshihiko Suyama Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, Fukuoka 814-0180, Japan Email: [email protected] Received March 24, 2020; accepted August 10, 2020
Abstract
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 Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces. Keywords
conformally flat hypersurface, system of evolution equations, Guichard net, integrability condition,
surface in 3-sphere MSC(2010)
53B25, 53E40
Citation: Suyama Y. Generic conformally flat hypersurfaces and surfaces in 3-sphere. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-020-1761-9
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Introduction
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 S3 . Here, we say that a hypersurface is generic if it has distinct three principal curvatures at each point. The classification of conformally flat hypersurfaces in (n + 1)-dimensional space forms was determined by Cartan [5] for n > 4: a hypersurface in an (n+1)-dimensional space form is conformally flat if and only if it is a branched channel hypersurface. Now, although 3-dimensional branched channel hypersurfaces are conformally flat as well, in the case of n = 3 there are also generic 3-dimensional conformally flat hypersurfaces. Our theme of this paper is to study these generic 3-dimensional conformally flat hypersurfaces: we relate generic conformally flat hypersurfaces to analytic surfaces in S3 . c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Suyama Y
Sci China Math
Any generic conformally flat hypersurface in a 4-dimensional space form has a special principal curvature line coordinate system (x, y, z): its first fundamental form I is expressed as I = l
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