Recurrent real hypersurfaces in complex two-plane Grassmannians
Let \(G_2({\mathbb C}^{n+2})\) denotes the set of all complex two-dimensional linear subspaces in \({\mathbb C}^{n+2}\) and let \(M\) be a real hypersurface in \(G_2({\mathbb C}^{n+2})\). Such a hypersurface admits the 3-dimensional distribution \({\mathfrak D}^{\bot}\), which is spanned by almost contact 3-structure vector fields \(\{\xi_1,\xi_2,\xi_3\}\), such that \(T_x M=\mathfrak D\oplus {\mathfrak D}^{\bot}\). When the shape operator \(A\) of a real hypersurface \(M\) is recurrent, i.e. \((\nabla_x A)Y=\beta(X)AY\) for a 1-form \(\beta\), then \(M\) is said to be a recurrent real hypersurface. The author proves the following non-existence result: Theorem. There do not exist any recurrent real hypersurfaces in \(G_2({\mathbb C}^{n+2})\), \(m\geqslant 3\), with \(\mathfrak D\) (resp. \({\mathfrak D}^{\bot}\))-invariant shape operator.
scientific article; zbMATH DE number 5082278