Almost Conformal $\eta$-Ricci Solitons in Three-Dimensional Lorentzian Concircular Structures

Abstract

The object of the present paper is to study the properties of three-dimensional Lorentzian concircular structure ($(LCS)_{3}$-)manifolds admitting the almost conformal $\eta$-Ricci solitons and gradient shrinking $\eta$-Ricci solitons. It is proved that an $(LCS)_3$-manifold with either an almost conformal $\eta$-Ricci soliton or a gradient shrinking $\eta$-Ricci soliton is a quasi-Einstein manifold. Also, the example of an almost conformal $\eta$-Ricci soliton in an $(LCS)_{3}$-manifold is provided in the region where $(LCS)_{3}$-manifold is expanding.