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

Abstract

We study four-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 four-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.

Keywords

• almost-hermitian metrics
• chern-einstein metrics
• hermitian connections
• weyl connection
• 53c55 (primary)
• 53b35 (secondary)