Ricci solitons in three-dimensional paracontact geometry

TL;DR

This paper classifies three-dimensional paracontact metric manifolds with Ricci soliton Reeb vector fields, revealing nontrivial examples unlike the contact case, and corrects previous results in the field.

Contribution

It provides a complete description of three-dimensional paracontact Ricci solitons, including explicit examples and a correction to earlier research findings.

Findings

Existence of nontrivial three-dimensional paracontact Ricci solitons.

Explicit homogeneous and inhomogeneous examples provided.

Correction of previous main results on normal paracontact Ricci solitons.

Abstract

We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentz\-ian) Ricci solitons are necessarily trivial, that is, $K$-contact and Einstein, the paracontact metric case allows nontrivial examples. Both homogeneous and inhomogeneous nontrivial three-dimensional examples are explicitly described. Finally, we correct the main result of [AGAG-D-13-00189], concerning three-dimensional normal paracontact Ricci solitons.