On Kähler-like and G-Kähler-like almost Hermitian manifolds

Abstract We introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B i¯j¯λBλji≥0 B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0 , then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.