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Fundamental theorem of geometry without the surjective assumption


Authors: Baokui Li and Yuefei Wang
Journal: Trans. Amer. Math. Soc. 368 (2016), 6819-6834
MSC (2010): Primary 51F15, 30C35
DOI: https://doi.org/10.1090/tran/6533
Published electronically: February 2, 2016
MathSciNet review: 3471078
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Abstract: In this paper, we solve the rigidity problem on geodesic maps in the hyperbolic space. The main result is that a geodesic-to-geodesic injection in hyperbolic space $ \mathbb{D}^n$ is an isometry or a composition of an isometry and an affine transformation under the Klein model if and only if it is non-degenerate. We first solve the rigidity problems on Euclidean space and the $ n$-sphere and show that a line-to-line injection in Euclidean space $ \mathbb{R}^n$ is an affine transformation if and only if it is non-degenerate and that a circle-to-circle injection on the $ n$-sphere $ \hat {\mathbb{R}}^n$ is a Möbius transformation if and only if it is non-degenerate. More general results for hyperplane-to-hyperplane maps are obtained.

The key method is to establish a new version of the celebrated Pappas' Theorem.


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Additional Information

Baokui Li
Affiliation: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China – and – Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
Email: henan_lbk@bit.edu.cn

Yuefei Wang
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: wangyf@math.ac.cn

DOI: https://doi.org/10.1090/tran/6533
Keywords: Line-to-line transformations, Pappus' Theorem, $g$-reflections, affine transformations, M\"obius transformations
Received by editor(s): August 20, 2013
Received by editor(s) in revised form: November 17, 2013, and June 10, 2014
Published electronically: February 2, 2016
Additional Notes: The first author was supported in part by the NSF of China (No. 11101032). The second author was supported in part by the NSF of China (No. 10831004)
Article copyright: © Copyright 2016 American Mathematical Society

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