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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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