$$*$$-$$\eta $$-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 $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> - $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> -Ricci soliton with the weak $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -Kenmotsu structure is studied and new characteristics of Einstein metrics are obtained.