| Summary: | In this paper, we study the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-Kenmotsu manifolds if its metrics are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>*</mo><mi>η</mi></mrow></semantics></math></inline-formula>-Ricci-Yamabe solitons. It is proven that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-Kenmotsu manifold endowed with a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>*</mo><mi>η</mi></mrow></semantics></math></inline-formula>-Ricci-Yamabe soliton is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Einstein. The necessary conditions for an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-Kenmotsu manifold, whose metric is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>*</mo><mi>η</mi></mrow></semantics></math></inline-formula>-Ricci-Yamabe soliton, to be an Einstein manifold are derived. Finally, we model an indefinite Kenmotsu manifold example of dimension 5 to examine the existence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>*</mo><mi>η</mi></mrow></semantics></math></inline-formula>-Ricci-Yamabe solitons.
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