On Ricci curvature of metric structures on $\mathfrak{g}$-manifolds

TL;DR

This paper investigates Ricci curvature properties of ${rak{g}}$-manifolds, especially in higher dimensions with abelian Lie algebras, exploring conditions for Ricci solitons and extending Einstein K-manifold results.

Contribution

It provides new conditions under which ${rak{g}}$-manifolds can be Ricci or gradient Ricci solitons and generalizes the Boyer-Galicki theorem to higher dimensions for abelian cases.

Findings

Identified conditions for Ricci and gradient Ricci solitons on ${rak{g}}$-manifolds.

Established a non-existence result for certain Einstein ${rak{g}}$-manifolds in higher dimensions.

Extended the Boyer-Galicki theorem to a special class of abelian ${rak{g}}$-manifolds.

Abstract

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the ${\mathfrak{g}}$-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian ${\mathfrak{g}}$-manifolds.