Canonical metric connections with constant holomorphic sectional curvature

TL;DR

This paper investigates the conjecture by Chen and Nie regarding space forms for canonical metric connections on compact Hermitian manifolds, verifying it for specific classes like complex nilmanifolds and non-balanced Bismut torsion-parallel manifolds.

Contribution

It confirms the Chen-Nie conjecture for particular Hermitian manifolds, expanding understanding of their geometric structures and curvature properties.

Findings

Verified the conjecture for complex nilmanifolds with nilpotent J

Confirmed the conjecture for non-balanced Bismut torsion-parallel manifolds

Enhanced understanding of space forms in Hermitian geometry

Abstract

We consider the conjecture of Chen and Nie concerning the space forms for canonical metric connections of compact Hermitian manifolds. We verify the conjecture for two special types of Hermitian manifolds: complex nilmanifolds with nilpotent $J$, and non-balanced Bismut torsion-parallel manifolds.