Orthogonal almost complex structures of hypersurfaces of purely imaginary octonions (Q614285)

In the first part of the paper the authors provide a new elementary proof of the structure equations of the exceptional Lie group \(G_2\), obtained earlier by \textit{E. Calabi} [Trans. Am. Math. Soc. 87, 407--438 (1958; Zbl 0080.37601)] and \textit{R. L. Bryant} [J. Differ. Geom. 17, 185--232 (1982; Zbl 0526.53055)]. Next, they define almost complex structures on hypersurfaces of the space of purely imaginary octonions, and give some properties. Then, they give a \(G_2\)-congruence theorem of Im\(\;\mathfrak{C}\) of hypersurfaces of Im\(\;\mathfrak{C}\). It is also shown that the Stiefel manifold of oriented \(2\)-frames in Im\(\;\mathfrak{C}\) is diffeomorphic to \(G_2/SU(2)\). Finally, they classify also complex structures of homogeneous hypersurfaces of Im\(\;\mathfrak{C}\) into \(4\) types.