Fundamental theorem of geometry without the surjective assumption

Li, Baokui; Wang, Yuefei

doi:10.1090/tran/6533

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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