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Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator. (English) Zbl 1058.53046
Let $$G_2(\mathbb{C}^{m+2})$$ be the Grassmann manifold of all 2-dimensional complex subspaces of $$\mathbb{C}^{m+2}$$. $$G_2(\mathbb{C}^{m+2})$$ is a compact irreducible Kählerian manifold, which is also a quaternionic Kähler but not a hyper-Kähler manifold. Let $$M\subset G_2(\mathbb{C}^{m+2})$$ be a real hypersurface. Let $$(g, J)$$ be the Kählerian structure of $$G_2(\mathbb{C}^{m+2})$$. $$(g, J)$$ induces an almost contact metric (a. ct. m.) structure $$(\varphi,\xi,\eta,g)$$ on $$M$$ [D. E. Blair, Contact manifolds in Riemannian geometry, Springer, Berlin) (1976; Zbl 0319.53026)]. Let $${\mathcal J}$$ be the quaternionic Kähler structure of $$G_2(\mathbb{C}^{m+2})$$ (as known, $${\mathcal J}$$ doesn’t contain $$J$$). Let $$\{J_\nu:\nu\in \{1,2,3\}\}$$ be a local frame of $${\mathcal J}$$. Each $$(g, J_\nu)$$ induces an a. ct. m. structure $$(\varphi_\nu, \xi_\nu, \eta_nu,\;g)$$ on $$M$$. Let $$A$$ be the Weingarten operator of the given immersion, corresponding to a fixed choice of unit normal. The author proves the two following results:
(1) There are no real hypersurfaces $$M$$ in $$G_2(\mathbb{C}^{m+2})$$ such that $$A\varphi_\nu= \varphi_\nu A$$, $$\nu\in \{1,2,3\}$$.
(2) Let $$M$$ be a Hopf real hypersurface $$M\subset G_2(\mathbb{C}^{m+2})$$ such that $$[A,\varphi_\nu]= 0$$ on $$(\mathbb{R}\xi)^\perp$$ $$(\nu\in \{1,2,3\})$$. Then $$M$$ is an open piece of a tube around $$\mathbb{Q} P^n$$ (immersed in $$G_2(\mathbb{C}^{m+2})$$ as a totally geodesic submanifold $$(m= 2n)$$).

##### MSC:
 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Zbl 0319.53026
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##### References:
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