Quaternion CR-submanifolds of a quaternion Kähler manifold
The quaternionic Kähler manifold \(\widetilde M\) was introduced and studied by \textit{S. Ishihara} [J. Differ. Geom. 9, 483-500 (1974; Zbl 0297.53014)], the quaternionic CR-submanifold \(M\) of such a manifold by \textit{M. Barros}, \textit{B. Y. Chen} and \textit{F. Urgano} [Kodai Math. J. 4, 399-417 (1981; Zbl 0481.53046)]. A local basis of almost Hermitian structures is denoted by \(\psi_1=I\), \(\psi_2=J\), \(\psi_3=K\). For any \(X\in TM\) and \(N\in T^\perp M\) there are decompositions into tangential and normal components \(\psi_r X=P_rX+ Q_rX\), \(\psi_rN= t_rN+f_rN\), where \(r=1,2,3\). The covariant differentiations are defined by \((\widetilde\nabla P_r)(Y)= \nabla(P_rY) -P_r \nabla_XY\) etc. If \(\widetilde\nabla P_r=0\) the endomorphism \(P_r\) is said to be parallel. It is proved that 1) \(Q_r\) is parallel if and only if \(t_r\) is parallel, 2) \(M\) is a QR-product (i.e. is locally the Riemannian product of a quaternionic submanifold and a totally real submanifold) if and only if \(P_r\) is parallel, 3) if \(D\) and \(D^\perp\) are orthogonal distributions on \(M\) and \(D^\perp\) is a totally real foliation then the Bott connection of \(D^\perp\) preserves the volume form of \(D\). A classification theorem is deduced for the totally umbilical quaternionic CR-submanifolds of a quaternionic Kähler manifold.
scientific article; zbMATH DE number 1751228