An Equivalence of Scalar Curvatures on Hermitian Manifolds

Abstract

For a Kähler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kähler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu-Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kähler. However, we prove that a certain class of noncompact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

Scalar curvatures in almost Hermitian geometry and some applications

“…(1) If t ≥ Recently, for the Chern connection D 1 , M. G. Dabkowski and M. T. Lock [11] have constructed non-compact Hermitian manifolds with 2s 1 (1) = s which are not Kähler manifolds. M. Lejmi and M. Upmeier have given a problem in Remark 3.3 in [27]: do higher-dimensional compact almost Hermitian non-Kähler manifolds with 2s 1 (1) = s exist?…”

Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

“…When M is a closed Hermitian manifold (the integrable case), one can deduce Theorem 3.2 in any dimension (see [22,35] or apply the technique above to Remark 2.4). On the other hand, Dabkowski-Lock [16] have examples of noncompact Hermitian with s H = s g which are not Kähler. Do higher-dimensional closed almost Hermitian non-Kähler manifolds with s H = s g exist?…”

Scalar curvatures of invariant almost Hermitian structures on flag manifolds with two and three isotropic summands

“…where ω n is the volume of the unity ball in R n , see [4], [6]. Therefore the Riemannian scalar curvature s(p) is positive or negative at a point p, if the volume of a small geodesic ball at p is respectively smaller or larger than the corresponding Euclidean ball of the same radius.…”