Almost Cosymplectic $$(k,\mu )$$-metrics as $$\eta$$-Ricci Solitons

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k<0$$ k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k<0$$ k < 0 . In addition, a concrete example is constructed to illustrate the above result.

Some Results on Cosymplectic Manifolds Admitting Certain Vector Fields

The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.

Identities of the Riemannian Curvature Tensor of Almost C(ʎ)-Manifolds

The geometry of the Riemannian curvature tensor of an almost C(λ)-manifold is studied. We have obtained several identities of the Riemannian curvature tensor of almost C(λ)-manifolds. Four additional identities are distinguished from these identities, on the basis of which four classes of almost C(λ)-manifolds are determined. A local classification of each of the distinguished classes of almost C(λ)-manifolds is obtained. It is proved that the set of almost C(λ)-manifolds of class R_1 coincides with the set of almost C(λ)-manifolds of class R_2, and it is also proved that the set of almost C(λ)-manifolds of class R_3 coincides with the set of almost C(λ)- manifolds of class R_4. We have found that an almost C(λ)-manifold, dimension greater than 3, is a manifold of class R_4 if and only if it is a cosymplectic manifold, i.e. when it is locally equivalent to the product of the Kähler manifold and the real line.

On Three Dimensional Cosymplectic Manifolds Admitting Almost Ricci Solitons

In the present paper we study three dimensional cosymplectic manifolds admitting almost Ricci solitons. Among others we prove that in a three dimensional compact orientable cosymplectic manifold M^3 withoutboundary an almost Ricci soliton reduces to Ricci soliton under certain restriction on the potential function lambda. As a consequence we obtain several corollaries. Moreover we study gradient almost Ricci solitons.

A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products.