Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds

<abstract><p>The present paper is to deliberate the class of $ \epsilon $-Kenmotsu manifolds which admits conformal $ \eta $-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal $ \eta $-Ricci soliton of $ \epsilon $-Kenmotsu manifolds. Moving further, we investigate some curvature conditions admitting conformal $ \eta $-Ricci solitons on $ \epsilon $-Kenmotsu manifolds. Next, we consider gradient conformal $ \eta $-Ricci solitons and we present a characterization of the potential function. Finally, we develop an illustrative example for the existence of conformal $ \eta $-Ricci soliton on $ \epsilon $-Kenmotsu manifold.</p></abstract>

η-*-Ricci Solitons and Almost co-Kähler Manifolds

The subject of the present paper is the investigation of a new type of solitons, called η-*-Ricci solitons in (k,μ)-almost co-Kähler manifold (briefly, ackm), which generalizes the notion of the η-Ricci soliton introduced by Cho and Kimura . First, the expression of the *-Ricci tensor on ackm is obtained. Additionally, we classify the η-*-Ricci solitons in (k,μ)-ackms. Next, we investigate (k,μ)-ackms admitting gradient η-*-Ricci solitons. Finally, we construct two examples to illustrate our results.

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.

Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.