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Hypersurfaces of the homogeneous nearly Kähler \(\mathbb{S}^6\) and \(\mathbb{S}^3\times\mathbb{S}^3\) with anticommutative structure tensors. (English) Zbl 1433.53028

Summary: Each hypersurface of a nearly Kähler manifold is naturally equipped with two tensor fields of \((1,1)\)-type, namely the shape operator \(A\) and the induced almost contact structure \(\phi \). In this paper, we show that, in the homogeneous nearly Kähler \(\mathbb{S}^6\) a hypersurface satisfies the condition \(A\phi+\phi A=0\) if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kähler manifold \(\mathbb{S}^3\times\mathbb{S}^3\) does not admit a hypersurface that satisfies the condition \(A\phi+\phi A=0\).

MSC:

53B25 Local submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
53C30 Differential geometry of homogeneous manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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