Eine Beckman-Quarles-Aussage für reguläre Metriken von Index \(\leq 2\). (A result of Beckman-Quarles type for regular metrics of index \(\leq 2.)\) (Q580712)

The paper is devoted to a generalization of a result proved by \textit{J. A. Lester} [Arch. Math. 37, 561-568 (1981; Zbl 0457.51027)], regarding the pseudo-metrics of \({\mathbb{R}}^ n\) induced by a form \(\Phi (x,y)=x\cdot y=\sum^{j}_{\mu =1}x_{\mu}y_{\mu}-\sum^{k}_{\nu =1}x_{j+\nu}y_{j+\nu},\) where the pseudo-metric is given by \(d(x,y)=(x-y)^ 2\), \((x,y\in {\mathbb{R}}^ n)\). Under the assumptions \(k+j=n\), \(n\geq 3\), \(j\geq 2\), \(k\geq 1\), the following important result is proved: Let \(f: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) be an application that keeps fixed the origin and such that \(\forall x\), \(y\in {\mathbb{R}}^ n\) \(d(x,y)=1 \Rightarrow d(f(x),f(y))=1\). Then f is a linear isometry. This result generalizes the one obtained by Lester for the case \(n\geq 3\), \(j=n-1,\) \(k=1.\) \textit{F. S. Beckman} and \textit{D. A. Quarles} jun. [Proc. Am. Math. Soc. 4, 810-815 (1953; 52, 182)] proved the preliminar result for the case \(n\geq 2\), \(j=n\), \(k=0\). Later \textit{W. Benz} [Math. Z. 177, 101-106 (1981; Zbl 0438.51015), Arch. Math. 34, 550-559 (1980; Zbl 0446.51015)] studied the case \(n\geq 2\), \(j=1\), \(k=n-1\), related to the transformations of Lorentz.