Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere
type: Item
instance of: scholarly article
title: Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere
zbMATH Open document ID: 0842.53015
DOI: 10.1007/BF01263564
author: John Bolton
author: Lyndon M. Woodward
author: Jürgen Berndt
published in: Geometriae Dedicata
publication date: 1996-07-28T00:00:00Z
review text: Using Cayley numbers, it is possible to define a vector cross product \(\times\) on \(\mathbb{R}^7\) and an almost complex structure \(J\) on \(S^6\). This almost complex structure is compatible with the standard metric \(g\) on \(S^6\). Special submanifolds, for example almost complex submanifolds (\(J\) maps the tangent space into the tangent space) or totally real submanifolds (\(J\) maps the tangent space into the normal space) have been studied by various authors, see a. o. [\textit{N. Ejiri}, Proc. Am. Math. Soc. 83, 759-763 (1981; Zbl 0474.53051)], [\textit{F. Dillen}, \textit{L. Verstraelen} and the reviewer, Bull. Am. Math. Soc., New Ser. 19, No. 2, 433-438 (1988; Zbl 0655.53052)] and the references contained therein. In the paper under review, Hopf hypersurfaces are studied. These are hypersurfaces \(M\) in \(S^6\) for which the integral curves of \(U= - J\xi\), \(\xi\) the unit normal of \(M\) in \(S^6\), are geodesics in \(M\). The main result is the following: Let \(M\) be a Hopf hypersurface in \(S^6\). Then \(M\) is an open part of either a geodesic hypersphere of \(S^6\) or a tube around an almost complex curve in \(S^6\). The above theorem gives yet another relation between two special classes of submanifolds of the nearly Kähler 6-sphere and illustrates the nice geometry which appears when studying those submanifolds in \(S^6\).
reviewed by: Luc Vrancken
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 806260
zbMATH Keywords: almost complex curves
zbMATH Keywords: nearly Kähler six-sphere
zbMATH Keywords: Hopf hypersurfaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 806260