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 Bi¯j¯λBλji≥0B_{\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.

Keywords

• almost hermitian manifolds
• kähler-like metrics
• g-kähler-like metrics
• chern connection
• 53c15 (primary)
• 53c55 (secondary)