Abstract
AbstractWe 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_{\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.
Highlights
The almost Hermitian geometry has been studied vigorously in last years such as in [14], [15], [16], [24] and [26]
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
Summary
The almost Hermitian geometry has been studied vigorously in last years such as in [14], [15], [16], [24] and [26]. In the Hermitian case, Yang and Zheng examined the Hermitian curvature tensors of Hermitian metrics, as the curvature tensors satis es all the symmetry conditions of the curvature tensor of a Kähler metric in [23] When a manifold is compact, these metrics are more special than balanced metrics since they are always balanced, that is, d(ωn− ) = , where ω is the fundamental -form associated to a Hermitian metric and n is the complex dimension of the manifold This fact has attracted attention in the reserch of non-Kähler Calabi-Yau manifolds. Yang and Zheng showed that when R = RL, g is Kähler in [23, Theorem 1.1], and they showed that when the manifold is compact, either condition, the Kähler-likeness or the G-Kähler-likeness, would imply that the metric is balanced. If it is either Kähler-like or G-Kähler-like, it must be balanced
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