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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 \(m\geq 3\).

MSC:
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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