Weak \(\beta \) -Kenmotsu Manifolds and \(\eta \) -Ricci Solitons

Weak contact metric structures on a smooth manifold have been recently introduced by V. Rovenski and R. Wolak in 2022. In this paper, we define a new structure of this kind called a weak \(\beta \) -Kenmotsu structure (that generalizes the notion by K. Kenmotsu with \(\beta =1\) and its extension for \(\beta \ne 0\) by Z. Olszak). We show that a weak \(\beta \) -Kenmotsu manifold is locally the warped product \((-\varepsilon ,\varepsilon )\times _\sigma \bar{M}\) , where \((\partial _t\,\sigma )/\sigma =\beta \ne 0\) , and \((\bar{M},\bar{g})\) is equipped with a parallel skew-symmetric (1,1)-tensor \(\bar{\phi }\) such that \(\bar{\phi }^{\,2}\) is negative definite. Then, we show that an \(\eta \) -Einstein weak \(\beta \) -Kenmotsu manifold with \(\beta =const\ne 0\) admitting an \(\eta \) -Ricci soliton structure is an Einstein manifold. Finally, we prove that a weak \(\beta \) -Kenmotsu manifold and admitting an \(\eta \) -Ricci soliton structure, whose non-zero potential vector field is weak contact or is collinear to \(\xi \) , is an Einstein manifold.