Fundamental theorem of geometry without the surjective assumption
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- by Baokui Li and Yuefei Wang PDF
- Trans. Amer. Math. Soc. 368 (2016), 6819-6834 Request permission
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
- 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)
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6819-6834
- MSC (2010): Primary 51F15, 30C35
- DOI: https://doi.org/10.1090/tran/6533
- MathSciNet review: 3471078