A generalization of the Schwarz lemma for transversally harmonic maps

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.