Fundamental theorem of hyperbolic geometry without the injectivity assumption

arXiv:0810.1580v2 [math.CV] 16 Feb 2009 Fundamental theorem of hyperbolic geometry without the injectivity assumption GUOWU YAO Department of Mathematical Sciences, Tsinghua University Beijing, 100084, P.R. China e-mail: gwyao@math.tsinghua.edu.cn 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. 1 . Introduction Let ˆRn = Rn ∪{∞} where Rn is the n−dimensional Euclidean space and let Hn be the n−dimensional hyperbolic space. A map f of ˆRn to itself is called r−sphere preserving if f maps an r−dimensional sphere onto an r−dimensional sphere. Similarly, a map f of Rn (or Hn) to Rn (or Hn) is called r−hyperplane preserving if f maps an r−di- mensional hyperplane onto an r−dimensional hyperplane in Rn (or Hn). In particular, when r = 1, we call the corresponding map f to be a circle-preserving (line-preserving, geodesic-preserving) map in ˆRn (Rn, Hn), respectively. In the sequel, we prescribe n ≥2 and 1 ≤r < n. The property of a M¨obius transformation acting on ˆC is so clear and the relations between M¨obius transformation and some of its property have been extensively studied. For examples, Carath´eodory first proved that if f : ˆC →ˆC is a circle-preserving bijection, then f is a M¨obius transformation (see [4] or [12]); Nehari [11] showed that if f : ˆC → ˆC is a non-constant meromorphic function that preserves circles, then f is a M¨obius transformation. Of course, the analogous problem for affine (or isometric) transformations on Rn (or Hn) is also concerned. In [7], Jeffers obtained the following extension of Carath´eodory’s 2000 Mathematics Subject Classification. Primary 37B05, 30C35; Secondary 51F15. Key words and phrases. M¨obius transformation, affine transformation, isometric transformation. The author was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) and the National Natural Science Foundation of China. 1 2 GUOWU YAO result to all three cases (for concision, we combine three theorems obtained by him into one). Theorem A. Suppose that f : ˆRn →ˆRn (Rn →Rn, Hn →Hn) is a bijection that preserves r−dimensional spheres (r−dimensional hyperplanes). Then f is a(n) M¨obius (affine, isometric) transformation. An r−sphere preserving map f is called degenerate if its image f(ˆRn) is an r−di- mensional sphere; otherwise, f is called non-degenerate. The reader will easily guess the proper definitions for non-degenerate and degenerate maps in the Euclidean and hyperbolic settings. In a recent article [8], B. Li and Y. Wang made neither injectivity nor surjectivity assumptions on f and proved Theorem B. Suppose that f : ˆRn →ˆRn (Rn →Rn, Hn →Hn) is a circle-preserving (line-preserving, geodesic-preserving) map. Then f is a(n) M¨obius (affine, isometric) transformation if and only if f is non-degenerate. The existence of degenerate maps was shown in [8, 14]. More recently, the author joint with B. Li [9] obtained the following generalization of Theorem B. Theorem C. Suppose that f : ˆRn →ˆRn (Rn →Rn, Hn →Hn) is an r−sphere preserving (r−hyperplane preserving) map. Then f is a(n) M¨obius (affine, isometric) transformation if and only if f is non-degenerate. In [5], Chubarev and Pinelis showed, among other things, that the injective condition for the Euclidean case Rn in Theorem A can be removed. Precisely, the following theorem was implied. Theorem D. Suppose that f : Rn →Rn is a surjective map and maps every r−dimen- sional hyperplane into an r−dimensional hyperplane. Then f is an affine transformation. Inspired by Theorem D, the following two conjectures were naturally posed in [9]: Conjecture 1. Suppose that f : ˆRn →ˆRn is a surjective map and maps every r−dimen- sional sphere into an r−dimensional sphere. Then f is a M¨obius transformation. and Conjecture 2. Suppose that f : Hn →Hn is a surjective map and maps every r−dimen- sional hyperplane into an r−dimensional hyperplane. Then f is an isometric transforma- tion. The aim of this paper is to prove Conjecture 2 by applying Theorem C but leave Conjecture 1 open. For completeness, we also give a simple proof of Theorem D in Section 5. Other results in the line can be found in [1, 2, 5, 6, 10, 13]. Remark 1. Recently, the author [15] proved that Conjecture 1 is true in the case r = n−1. Fundamental theorem of hyperbolic geometry 3 2 . Some preparations This section is devoted to reduce the proof of Conjecture 2 to that of the special case when r = 1. That is, we only n

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