QR-submanifolds and almost contact 3-structure (Q2703115)

Let \((\overline M,V,g)\) be an quaternionic Kählerian manifold, i.e., a Riemann manifold with metric \(g\), which admits a 3-dimensional vector bundle \(V\) of tensors of type (1,1) on \(\overline M\), such that in any coordinate neighborhood of \(\overline M\) there are local almost Hermitian structures \(J_1\), \(J_2\), \(J_3\), which satisfy \(J_1J_2= -J_2J_1=J_3\) and well-known conditions with respect to the Levi-Civita connection. A submanifold \(M\) isometrically immersed in \(\overline M\) is called a quaternionic-real submanifold (QR-submanifold) if there exists a vector subbundle \(\nu\) of the normal bundle \(TM^\perp\) such that \(J_a(\nu)=\nu\) and \(J_a (\nu^\perp) \subset T(M)\), for \(a=1,2,3\) [see \textit{A. Bejancu}, Geometry of CR-Submanifolds, Mathematics and its Applications, Reidel (1986; Zbl 0605.53001)]. In this paper the authors prove that if \(M\) is a QR-submanifold with \(\dim\nu^\perp=1\) then it admits an almost contact 3-structure. They give conditions for the almost contact 3-structure to be cosymplectic, Sasakian or nearly Sasakian and also obtain some results on the quaternion distribution.