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, Riemannian dual of Lee form is Killing vector field. use observation to describe locally conformally symplectic manifolds, and give some examples such Finally, problem on unimodular almost-abelian Lie algebras, classifying those admit left-invariant metric with parallel non-zero form.