Hermitian Manifolds with Flat Associated Connection

TL;DR

This paper classifies Hermitian manifolds with flat associated connections, characterizing those with locally conformal metrics and relating them to locally conformal Kähler manifolds and surfaces with specific curvature properties.

Contribution

It provides a local classification of Hermitian manifolds with flat associated connections and characterizes those with conformal metrics and specific curvature conditions.

Findings

Hermitian manifolds with flat associated connection are classified locally.

Locally conformal Kähler manifolds are characterized by a curvature identity.

Hermitian surfaces with vanishing associated conformal curvature tensor are identified.

Abstract

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kaehler manifold of constant holomorphic sectional curvatures.