Second order parallel tensors and Ricci solitons in $$3$$ 3 -dimensional normal paracontact geometry

Abstract

The aim of this paper is to study two problems in the framework of paracontact geometry of dimension \(3\), namely, the class of parallel symmetric tensor fields of \((0, 2)\)-type and possible Lorentz Ricci solitons. We search for two types of Ricci solitons: the first when its potential vector field is exactly the characteristic vector field \(\xi \) of the paracontact structure and the second when the potential vector field is a paracontact-holomorphic one. For the former case we find all variants of Ricci solitons, expanding, steady and shrinking, and the fact that \(\xi \) is a conformal Killing vector field. A class of examples is completely discussed.