Gradient Ricci solitons carrying a closed conformal vector field

TL;DR

This paper classifies complete gradient Ricci solitons with constant scalar curvature that admit a non-parallel closed conformal vector field, revealing their geometric structure and special cases like Ricci-flatness and flatness.

Contribution

It provides a classification of such Ricci solitons under specific geometric conditions, including the Kähler case and harmonic 1-form scenarios.

Findings

Complete gradient Ricci solitons with the given conditions are isometric to well-known spaces.

Kähler gradient Ricci solitons with these properties are Ricci-flat and flat in dimension 4.

Solitons with harmonic associated 1-form have constant scalar curvature.

Abstract

We show that a complete gradient Ricci soliton $(M^n,\,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a K\"ahler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar curvature.