On gradient generalized $\vartheta$-$\eta$-Ricci solitons with applications in space-times

Abstract

UDC 514.7; 530.12

We explore the generalized $\vartheta$-$\eta$-Ricci solitons on paracontact and $K$-paracontact metric manifolds, with focus on the gradient case. First, we show that if a paracontact manifold satisfying $Q\phi=\phi Q$ and a potential vector field is parallel to the Reeb vector field, then the manifold $M$ of dimension $2n+1$ is $\eta$-Einstein with scalar curvature equal to $2n(\lambda -1)$. Subsequently, for the gradient case, we observe some intriguing outcomes. Specifically, for a $K$-paracontact manifold corresponding to a gradient generalized $\vartheta$-$\eta$-Ricci soliton, the potential function is constant under a certain condition. Finally, we present some applications of the gradient generalized $\vartheta$-$\eta$-Ricci soliton within the framework of the general relativity.