$\ast$-$\eta$-Ricci solitons and Einstein metrics on a weak $\beta$-Kenmotsu manifold

Weak almost contact metric manifolds (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications. An important case of these manifolds, which is locally a twisted product, is a weak $\beta$-Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the $\ast$-Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $\ast$-$\eta$-Ricci soliton with the weak $\beta$-Kenmotsu structure (with $\beta =const$) is studied and new characteristics of Einstein metrics are obtained.