A generalization of Nehari's univalence criterion
type: Item
instance of: scholarly article
title: A generalization of Nehari's univalence criterion
zbMATH Open document ID: 0722.53029
DOI: 10.1007/BF02566604
author: Dennis Stowe
author: Brad G. Osgood
published in: Commentarii Mathematici Helvetici
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140192
review text: Continuing a theme of some previous work, the authors study a natural generalization of the Schwarzian derivative in the setting of conformal mappings between Riemannian manifolds of arbitrary dimensions. Specifically, given a conformal local diffeomorphism f between Riemannian manifolds, this generalization is a certain symmetric, traceless, (0,2) tensor field S(f) on the domains of f. The main result is as follows: Suppose that (M,g) is a Riemannian manifold of dimension \(n\geq 2\) where the scalar curvature is at most n(n-1)K for some real K and any two points of M can be joined by a geodesic of length at most \(\delta\) for some \(0<\delta \leq \infty\). If f is a conformal local diffeomorphism from (M,g) to the standard round sphere, then f is injective provided that the norm of S(f) (as a bilinear form on each tangent space) is at most \(2(\pi^ 2/\delta^ 2)-(K/2).\) This generalizes a well-known univalence criterion of Nehari for analytic functions on the unit disk.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 4190608
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: conformal mappings
zbMATH Keywords: conformal local diffeomorphism
zbMATH Keywords: round sphere
zbMATH Keywords: univalence criterion
MaRDI profile type: Publication
OpenAlex ID: W2150933005
description: scientific article; zbMATH DE number 4190608