On real hypersurfaces in quaternionic projective space with \({\mathcal D}^\perp\)-recurrent second fundamental tensor
Let \(M\) be a real hypersurface in quaternionic projective space \(QP^m\), \(m\geq 3\), and denote by \((\phi_i, \xi_i, \eta_i,g)\), \(i=1,2,3\), the local almost contact metric structures on \(M\) which are induced by some canonical local basis of the quaternionic Kähler structure of \(QP^m\). Moreover, denote by \(A\) the shape operator of \(M\), by \(\nabla\) the Levi Civita covariant derivative of \(M\), and by \(D\) the maximal quaternionic subbundle of \(TM\). The authors classify all real hypersurfaces in \(QP^m\) which satisfy \(g((\nabla_X A)Y,Z)= \alpha(X) g(AY,Z)\) for some one-form \(\alpha\) on \(D\) and \(g((A\phi_i- \phi_iA)X,Y)=0\) for all \(X,Y,Z\in D\). They show that such a hypersurface \(M\) is either congruent to a tube around some totally geodesic \(QP^k\subset QP^m\) for some \(k\in \{0,\dots, m-1\}\) or \(M\) is ruled.
scientific article; zbMATH DE number 1275564