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Real hypersurfaces of quaternionic projective space satisfying \(\nabla_{U_i}R=0\). (English) Zbl 0901.53011

Let \(M\) be a real hypersurface of quaternionic projective space \(\mathbb{H} P^m\), \(m\geq 3\), endowed with the Fubini-Study metric of constant quaternionic sectional curvature 4. By applying the almost Hermitian structures in the quaternionic Kähler structure of \(\mathbb{H} P^m\) to the normal bundle of \(M\), one obtains a 3-dimensional subbundle \(D\) of the tangent bundle \(TM\) of \(M\). Denote by \(R\) the Riemannian curvature tensor of \(M\). The authors prove that if \(\nabla_X R=0\) for all \(X\in D\), then \(M\) is an open part of a tube of radius \(\pi/4\) over a totally geodesic \(\mathbb{H} P^k\) in \(\mathbb{H} P^m\) for some \(k\in \{0, \dots, m-1\}\). It is worthwhile to mention that there are no real hypersurfaces with parallel Riemannian curvature tensor in \(\mathbb{H} P^m\).

MSC:

53B25 Local submanifolds
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References:

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