Year: 2026
Type: preprint
Abstract: A long-standing conjecture in non-Kähler geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature should be Kähler when the constant is nonzero and Chern flat when the constant is zero. In this article, we study the corresponding problem for a Hermitian connection $\nabla$ with $\nabla T=0$. We first show that if the $\nabla$-holomorphic sectional curvature is constant, then the $(1,1)$-part of the curvature is $\nabla$-parallel. Our main result states that a nonzero constant forces $T=0$; consequently, the metric is Kähler and locally a complex space form. The proof is pointwise and requires neither compactness nor completeness. We also obtain applications to the Chern and Bismut connections.
Source: arXiv (Cornell University)
Authors: Chen, Shuwen, Zheng, Fangyang
Institution: Chongqing Normal University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3604482852, G4952392286
OpenAlex ID: W7214699460