We characterize the Einstein metrics in such broad classes of as almost $$\eta $$ -Ricci solitons and on Kenmotsu manifolds, generalize some known results. First, we prove that a metric an soliton is if either it -Einstein or potential vector field V infinitesimal contact transformation collinear to Reeb field. Further, manifold admits gradient with leaving scalar curvature invariant, then mani...

The object of the present paper is to characterize class Kenmotsu manifolds which admits conformal $$\eta $$ -Ricci soliton. Here, we have investigated nature soliton within framework manifolds. It shown that an -Einstein manifold admitting Einstein one. Moving further, considered gradient on and established a relation between potential vector field Reeb field. Next, it proved under certain con...

A transformation of an n-dimensional Riemannian manifold M , which transforms every geodesic circle of M into a geodesic circle, is called a concircular transformation. A concircular transformation is always a conformal transformation. Here geodesic circle means a curve in M whose first curvature is constant and second curvature is identically zero. Thus, the geometry of concircular transformat...