A characterization of balanced manifolds

A Hermitian metric on a complex manifold is Kähler if and only if it approximates the Euclidean metric to order 2 at each point, in a suitable coordinate system. We prove here an analogous characterization of balanced metrics, namely, a Hermitian metric is balanced if and only if its fundamental form ω has closed trace and ωi, j (z) does not contain linear terms involving zi , z j , \bar zi , \bar z j , for each point, in a suitable coordinate system.