A study on conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric connection (Q415753)

A K-contact, respectively the Kenmotsu manifold, is a contact metric manifold satisfying an additional compatibility condition. The semi-symmetric metric connection on a contact metric manifold differs from the Levi-Civita connection by a skew-symmetric tensor of a specific form. (In particular, the metric is still parallel with respect to that connection.) The C-Bochner curvature tensor and the \(\eta\)-Einstein metric condition are adaptations of the corresponding classical notions to contact metric manifolds. The two results of the paper state that a K-contact, respectively a Kenmotsu manifold, whose C-Bochner curvature is conservative with respect to the semi-symmetric metric connection, is necessarily \(\eta\)-Einstein with respect to the Levi-Civita connection.NEWLINENEWLINENEWLINEMost of the paper is devoted to computations rather than to explanations. Two concrete examples of manifolds together with a discussion of their properties are appended.