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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.)
Citations:
Zbl 0319.53026
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