On the second fundamental tensor of real hypersurfaces in quaternionic hyperbolic space (Q1596565)

Let \({\mathbf Q}H^m\) \((m\geq 2)\) be a quaternionic hyperbolic space of constant quaternionic sectional curvature \(c=-4\) and with Levi-Civita connection \(\overline\nabla\). Let \(M\) be a smooth connected real hypersurface without boundary in \({\mathbf Q}H^m\) and \(N\) its local normal unit vector field. Then the Weingarten formula is \(\overline\nabla_XN=-AX\) for \(X\) tangent to \(M\) [see \textit{J. Berndt}, J. Reine Angew. Math. 419, 9-26 (1991; Zbl 0718.53017)]. The maximal quaternionic distribution of \(M\) is denoted by \({\mathbf D}\). It is shown that if \(A\varphi_kX=\varphi_kAX\) for any \(X\in {\mathbf D}\), \(k=1,2,3\), where \(\varphi_kX\) is the tangential component of \(J_kX\) for every quaternionic unit \(Jf_k\), then \(M\) is an open subset of one of the following: a) a tube of radius \(r>0\) over a totally geodesic \({\mathbf Q} H^p\), \((p=0,\dots, m-1)\), b) a horosphere. If for the metric tensors \(g\) and \(g^0\) on \({\mathbf D}\) there holds \(g^0=ag\) for a smooth function on \(M\), then it is shown that \(a\) is constant; moreover, a classification of \(M\) for this case is given. The same is done for \(M\) of constant quaternionic sectional curvature. By \textit{Y. Tashiro} and \textit{S. Tachibana} [Kodai Math. Sem. Rep. 15, 176-183 (1963; Zbl 0116.39001)] has been shown the nonexistence of real hypersurfaces in \({\mathbf Q}H^m\) with parallel second fundamental tensor. Now also the nonexistence of semiparallel real hypersurfaces in \({\mathbf Q}H^m\) is established.