Conformal geometry, Euler numbers, and global invertibility in higher dimensions
Differential Geometry
2020-03-02 v1 Algebraic Geometry
Analysis of PDEs
Algebraic Topology
Complex Variables
Abstract
It is shown that in dimension at least three a local diffeomorphism of Euclidean n-space into itself is injective provided that the pull-back of every plane is a Riemannian submanifold which is conformal to a plane. Using a similar technique one recovers the result that a polynomial local biholomorphism of complex -space into itself is invertible if and only if the pull-back of every complex line is a connected rational curve. These results are special cases of our main theorem, whose proof uses geometry, complex analysis, elliptic partial differential equations, and topology.
Cite
@article{arxiv.2002.12884,
title = {Conformal geometry, Euler numbers, and global invertibility in higher dimensions},
author = {Frederico Xavier},
journal= {arXiv preprint arXiv:2002.12884},
year = {2020}
}