Fundamental theorem of hyperbolic geometry without the injectivity assumption

2008/10/09 by Guowu Yao, Yao, Guowu · 1 citation
Mathematics · #30C35 #37B05 #51F75 #Complex Variables (math.CV) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Mathematics and Applications #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.0810.1580

openalex publication_date 2008/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℍⁿ be the n-dimensional hyperbolic space. It is well known that, if f: ℍⁿ→ ℍⁿ 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: ℍⁿ→ ℍⁿ 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.

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