In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci -flat with respect to another canonical connection if and only if it is cosymplectic (i.e. d * ω = 0).
introductionLet (M, g, J, ω) be an almost Hermitian manifold. Gauduchon introduced in [13] a 1-parameter family ∇ t of canonical Hermitian connections which can be distinguished by the properties of the torsion tensor T . In this family ∇ 1 corresponds to so-called Chern connection which can be defined as the unique Hermitian connection whose (1, 1)-part of the torsion vanishes. In the quasi-Kähler case (i.e. when ∂ω = 0), the line {∇ t } degenerates to a single point and the Chern connection is the unique canonical connection.Any canonical connection ∇ t induces the so-called Ricci form ρ t (X, Y ) = 2 i tr ω R t (X, Y ), where R t denotes the curvature of ∇ t . It turns out that ρ t is always a closed form which can be locally written as the derivative of the 1-form θ t (X) = n r=1 g(∇ t X Z r , Z r ), where {Z r } is a (local) unitary frame. Moreover, in the cosymplectic case (i.e. when dω n−1 = 0) the line {θ t } degenerates to a single point (see Corollary 3.3) and all the canonical connections have the same Ricci form.The aim of this paper is to study the Ricci forms ρ t on 2-step nilmanifolds equipped with a left-invariant almost Hermitian structure. We recall that by definition a k-step nilmanifold is a compact quotient of a k-step nilpotent Lie group G by lattice. Since we are considering left-invariant almost Hermitian structures, we can work on Lie algrebras in an algebraic fashion. Our main result is the following Theorem 1. Let (g, g, J, ω) be a 2-step nilpotent Lie algebra with an almost Hermitian structure. Then (g, J) is Ricci-flat with respect to the Chern connection and it is Ricci-flat with respect to another canonical connection if and only if it is cosymplectic (i.e. d * ω = 0). This theorem has the following immediate consequence: Corollary 1.1. Every left-invariant almost Hermitian structure on a nilmanifold associated to a 2-step Lie group is Ricci-flat with respect to the Chern connection.