Conformal Ricci Solitons and related Integrability Conditions

TL;DR

This paper introduces conformal Ricci solitons in Riemannian geometry, establishes integrability conditions, and explores their relation to known structures like Einstein manifolds and Ricci solitons.

Contribution

It defines conformal Ricci solitons, derives integrability conditions, and constructs new tensors related to these structures, extending previous results in the literature.

Findings

Derived necessary integrability conditions for conformal Ricci solitons.

Constructed new (0,3)-tensors related to geometric structures.

Established transformation laws under conformal changes.

Abstract

In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial $(0,3)$-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor $D$ recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.