Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere
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 \(S^6\).
scientific article; zbMATH DE number 806260
scientific article