Fundamental theorem of hyperbolic geometry without the injectivity assumption

TL;DR

This paper proves that a surjective map on hyperbolic space that maps hyperplanes to hyperplanes must be an isometry, removing the need for injectivity assumptions present in classical results.

Contribution

It establishes a new characterization of hyperbolic isometries without assuming injectivity or hyperplane preservation, extending classical theorems.

Findings

Surjective hyperplane-preserving maps are isometries

The proof differs from previous approaches in Euclidean space

The spherical case remains an open problem

Abstract

Let $\mathbb{H}^n$ be the $n-$dimensional hyperbolic space. It is well known that, if $f: \mathbb{H}^n\to \mathbb{H}^n$ is a bijection that preserves $r-$dimensional hyperplanes, then $f$ is an isometry. In this paper we make neither injectivity nor $r-$hyperplane preserving assumptions on $f$ and prove the following result: Suppose that $f: \mathbb{H}^n\to \mathbb{H}^n$ is a surjective map and maps an $r-$hyperplane into an $r-$hyperplane, then $f$ is an isometry. The Euclidean version was obtained by A. Chubarev and I. Pinelis in 1999 among other things. Our proof is essentially different from their and the similar problem arising in the spherical case is open.