De U. C.
Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 620–628
The aim of the present paper is to study 3-dimensional f-Kenmotsu manifolds and Ricci solitons. First, we give an example of a 3-dimensional f-Kenmotsu manifold. Then we consider a Riccisemisymmetric 3-dimensional f-Kenmotsu manifold and prove that a 3-dimensional f-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold. Moreover, we investigate an η-parallel Ricci tensor in a 3-dimensional f-Kenmotsu manifold. Finally, we study Ricci solitons in a 3-dimensional f-Kenmotsu manifold.
Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1330-1329
The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure tensor $\phi$ under certain conditions, and the operator $Q\phi - \phi Q$ is invariant under a $D$-homothetic deformation. We also discuss the invariance of $\eta$-Einstein manifolds, $\phi$-sectional curvature, and the local $\phi$-Ricci symmetry under a $D$-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete example.