A generalization of Nehari's univalence criterion
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.
scientific article; zbMATH DE number 4190608