A characterization of balanced manifolds

Abstract

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,zj,zi¯,zj¯, for each point, in a suitable coordinate system.