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

Abstract 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 in mathematics and physics. An important case of these manifolds, which is locally a twisted product, is a weak $$\beta $$ β -Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the $$*$$ ∗ -Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $$*$$ ∗ - $$\eta $$ η -Ricci soliton with the weak $$\beta $$ β -Kenmotsu structure is studied and new characteristics of Einstein metrics are obtained.