Second-Chern-Einstein metrics on 4-dimensional almost-Hermitian manifolds

TL;DR

This paper investigates 4-dimensional second-Chern-Einstein almost-Hermitian manifolds, characterizing their properties, especially in the compact case, and classifying certain Lie algebra examples with specific geometric structures.

Contribution

It provides new insights into the structure of second-Chern-Einstein manifolds, including classifications and examples in the compact and Lie algebra settings.

Findings

Riemannian dual of Lee form is a Killing vector under certain conditions

Classification of second-Chern-Einstein structures on unimodular almost-abelian Lie algebras

Examples of compact second-Chern-Einstein locally conformally symplectic manifolds

Abstract

We study 4-dimensional second-Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe 4-dimensional compact second-Chern-Einstein locally conformally symplectic manifolds and we give some examples of such manifolds. Finally, we study the second-Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second-Chern-Einstein metric with a parallel non-zero Lee form.