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 \(E\not \subset {\hat{\mathbb {R}}}\) be an arc or a circle. If a map \(f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}\) preserves cross-ratios in E, then f is a Möbius transformation when \({\bar{E}}\not =E\) and f is a Möbius or conjugate Möbius transformation when \({\bar{E}}=E\), where \({\bar{E}}=\{{\bar{z}}|z\in E\}\). (II) Let \(E\subset {\hat{\mathbb {R}}}\) be an arc satisfying the condition that the cardinal number \(\#(E\cap \{0,1,\infty \})<2\). If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption \(\#(E\cap \{0,1,\infty \})<2\) is necessary.

在本文中，我们对由 Beardon 和 Minda 提出的与交叉比相关的黎曼球面上映射的刚性问题给出了肯定的答案（Proc Am Math Soc 130(4):987–998, 2001）。我们的主要结果是： (I) 设\(E\not \subset {\hat{\mathbb {R}}}\)是一个弧或一个圆。如果映射\(f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}\)保留了E 中的交叉比，那么f是一个莫比乌斯变换，当\({ \bar{E}}\not =E\)并且f是当\({\bar{E}}=E\)时的莫比乌斯或共轭莫比乌斯变换，其中\({\bar{E}}=\{{ \bar{z}}|z\in E\}\)。（二）令\(E\subset {\hat{\mathbb {R}}}\)是满足基数\(\#(E\cap \{0,1,\infty \})<2\)条件的弧。如果f保留E 中的交叉比，则f是莫比乌斯或共轭莫比乌斯变换。提供的例子表明假设\(\#(E\cap \{0,1,\infty \})<2\)是必要的。