A generalization of the Schwarz lemma for transversally harmonic maps

TL;DR

This paper extends the Schwarz lemma to transversally harmonic and holomorphic maps between Riemannian and Kähler foliated manifolds, using Bochner techniques and sub-Laplacian comparison.

Contribution

It introduces a generalized Schwarz lemma for transversally harmonic maps with bounded dilatation and for transversally holomorphic maps between Kähler foliations.

Findings

Established a Schwarz lemma for transversally harmonic maps.

Derived a Schwarz type lemma for transversally holomorphic maps.

Utilized Bochner techniques and sub-Laplacian comparison in proofs.

Abstract

In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the Schwarz lemma for transversally harmonic maps of bounded generalized transversal dilatation. In addition, we also obtain a Schwarz type lemma for transversally holomorphic maps between K\"ahler foliations.