Baokui Li

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let $$E\not \subset {\hat{\mathbb {R}}}$$ E ⊄ R ^ be an arc or a circle. If a map $$f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}$$ f : C ^ ↦ C ^ preserves cross-ratios in E, then f is a Möbius transformation when $${\bar{E}}\not =E$$ E ¯ ≠ E and f is a Möbius or conjugate Möbius transformation when $${\bar{E}}=E$$ E ¯ = E , where $${\bar{E}}=\{{\bar{z}}|z\in E\}$$ E ¯ = { z ¯ | z ∈ E } . (II) Let $$E\subset {\hat{\mathbb {R}}}$$ E ⊂ R ^ be an arc satisfying the condition that the cardinal number $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 . If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 is necessary.

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to $\theta :(x_1,x_2,\dots ,x_n)\to \left({\displaystyle \frac{1}{x_1}},{\displaystyle \frac{x_2}{x_1}},\dots ,{\displaystyle \frac{x_n}{x_1}}\right).$ We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of ${\mathbb{R}}^n$ by using reflection-like maps. We mainly prove that a line-to-line map $\eta$ on any convex domain satisfying ${\eta }^{\circ 2}=id$ and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection $f:{\mathbb{D}}^n\mapsto {\mathbb{D}}^n$ is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

Recently, the first author and Y. Wang proved that $f\colon \rn\to\rn$ (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.