CR-submanifolds of a nearly trans-Sasakian manifold (Q1607931)

Let \(M\) be an \(m\)-dimensional isometrically immersed submanifold of a nearly trans-Sasakian manifold \(\overline M\). Definition. An \(m\)-dimensional Riemannian submanifold \(M\) of a nearly trans-Sasakian manifold \(\overline M\) is called a CR-submanifold if \(\xi\) is a vector field tangent to \(M\) and if there exists a differentiable distribution \(D:x\in M\to D_x \subset T_xM\) such that 1. \(D\) is invariant under \(\Phi\) that is \(\Phi D_x \subset D_x\) for each \(x\in M\); 2. The complementary orthogonal distribution \(D^\perp :x\in M\to D_x^\perp \subset T_xM\) of \(D\) is antiinvariant under \(\Phi\) that is, \(\Phi D_x^\perp \subset T_x^\perp M\) for all \(x\in M\), where \(T_xM\) and \(T_x^\perp M\) are the tangent space and the normal space of \(M\) at \(x\), respectively. In 1978 A. Bejancu introduced the notion of a CR-submanifold of a Kähler manifold. In 1985 J. A. Oubina introduced a new class of almost contact metric manifolds known as trans-Sasakian manifolds. This class contains \(\alpha\)-Sasakian and \(\beta\)-Kenmotsu manifolds. The geometry of CR-submanifold of trans-Sasakian manifold was studied by M. H. Shahid (1991) and (1994). A nearly trans-Sasakian is a more general concept (see A. Bejancu and N. Papaghiuc (1984)). The results of this note are the generalization of results obtained by several authors (M. Kobayashi (1981), M. H. Shahid (1994) and others).