The rigidity of almost Ricci-Yamabe solitons with concurrent-recurrent vector fields

Abstract

In this paper, we study almost Ricci-Yamabe soliton of $(\alpha, \beta)$-type which admitting concurrent-recurrent vector field. We show that if the potential vector field is (\romannumeral 1) conformal vector field, or (\romannumeral 2) pointwise collinear to the concurrent-recurrent vector field, or (\romannumeral 3) gradient of a smooth function, then the soliton is Einstein or almost $\beta$-Yamabe soliton. At the end of this paper, we give explicitly some manifolds which admit (or do not admit) concurrent-recurrent vector field.