Abstract
In this paper, we characterize three-dimensional Riemannian manifolds M(3 )admitting Ricci-Yamabe solitons (RYSs). It is proved that if an M-3 endowed with a semi-symmetric metric zeta-connection admits an RYS, then the scalar curvature of M-3 satisfies the Poisson equation delta r = 8(2 alpha-3 beta-rho)/beta , where alpha,beta is an element of Double-struck capital R and beta&NOTEQUexpressionL;0. We also discuss the existence of gradient RYS in Riemannian setting. Finally, we construct a nontrivial example of three-dimensional Riemannian 3-manifolds admitting RYS to prove some of our results.