Impact of Gradient $$\boldsymbol{\eta}$$-Ricci–Yamabe Solitons on Warped Product Manifolds with Applications

The main topic of this paper is to characterize gradient $$\eta$$ -Ricci–Yamabe solitons on warped product manifolds. At first, we establish necessary and sufficient conditions under which a warped product manifold admits gradient $$\eta$$ -Ricci–Yamabe solitons. Next, we deduce the condition under which a warped product manifold is conformal to a sphere in Euclidean space. Finally, we obtain some results in GRW space-times.