On the Conformal Factor in the Conformal Analogue of the Killing Equation on Cahen — Wallach Manifolds with Zero Weyl Tensor

Conformally Killing vector fields play an important role in the study of the group of conformal transformations of a manifold, Ricci flows on a manifold, and the theory of Ricci solitons. Lorentzian symmetric spaces are studied in detail in Lorentzian geometry and theoretical physics. These spaces are classified by Cahen and Wallach, and their properties are well studied in dimension 4 due to their various applications in physics. Killing vector fields and Ricci solitons on generalized Cahen — Wallach spaces were studied by D.N. Oskorbin, E.D. Rodionov and others. Killing vector fields allow finding a general solution to the Ricci soliton equation corresponding to the Einstein constant for the cases when the Einstein constant maintains its constancy. However, conformally Killing vector fields play the role of the Killing fields when the Einstein constant varies. It is known that the conformal factor in the conformal analogue of the Killing equation is constant for a nonzero Weyl tensor. In this paper, the conformal analogue of the Killing equation on Cahen — Wallach manifolds with a zero Weyl tensor is studied, and a general form of the conformal factor of this equation is obtained.