Schwarz lemma: the case of equality and an extension

TL;DR

This paper proves that equality in Schwarz lemmas implies global equality and total geodesicity of maps, and extends Schwarz lemma concepts to almost Hermitian manifolds using holomorphic sectional curvatures.

Contribution

It establishes that local equality in Schwarz lemmas implies global equality and extends Schwarz lemma results to almost Hermitian manifolds.

Findings

Equality at one point implies global equality in Schwarz lemmas.

Holomorphic maps are totally geodesic with constant rank under equality.

Schwarz lemma is extended to almost Hermitian manifolds using holomorphic sectional curvatures.

Abstract

We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.