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Real hypersurfaces in complex two-plane Grassmannians. (English) Zbl 0920.53016
The complex two-plane Grassmannian \(N:=G_2 (\mathbb{C}^{m+2})\) is a Riemannian symmetric space distinguished by the fact that it is equippped with a Kähler structure \(J\) and a quaternionic Kähler structure \({\mathfrak I}\) (which is a special parallel subbundle of \(\text{End} (TN)\) of rank 3). For any real hypersurface \(M\) of \(N\) then \(E_1:=J (\perp M)\) resp. \(E_3:={\mathfrak I} (\perp M)\) is a one- resp. three-dimensional subbundle of \(TM\). If \(M\) is an open part of one of the tubes around the canonically embedded \(G_2(\mathbb{C}^{m+1})\) or around the canonically embedded \(\mathbb{H} P^n\), then the subbundles \(E_1\) and \(E_3\) are invariant with respect to the shape operator of \(M\). The authors show that up to congruences there are no other hypersurfaces with the last property, if \(m\geq 3\).

53B25 Local submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C35 Differential geometry of symmetric spaces
53B35 Local differential geometry of Hermitian and Kählerian structures
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