On the boundedness of the sectional curvature of almost Hermitian manifolds

TL;DR

This paper establishes conditions under which almost Hermitian manifolds with bounded sectional curvature have constant holomorphic or antiholomorphic sectional curvature, extending known results to indefinite and definite metrics.

Contribution

It proves that bounded holomorphic or antiholomorphic sectional curvature implies pointwise constancy in almost Hermitian manifolds with indefinite or definite metrics.

Findings

Manifolds with bounded holomorphic curvature are of pointwise constant holomorphic curvature.

Manifolds with bounded antiholomorphic curvature are of pointwise constant antiholomorphic curvature.

Results apply to both indefinite and definite metric cases.

Abstract

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold is of pointwise constant antiholomorphic sectional curvature. Similar results are obtained for almost Hermitian manifolds of definite metric.