$G$-space of isotropic directions and $G$-spaces of $\phi$-scalars with $G=O(n,1,\mathbb{R})$

Misiak, Aleksander ; Stasiak, Eugeniusz

Mathematica Bohemica, Tome 133 (2008) no. 3, pp. 289-298.

Résumé

There exist exactly four homomorphisms $\phi$ from the pseudo-orthogonal group of index one $G=O(n,1,\mathbb{R})$ into the group of real numbers ${\mathbb{R}}_0.$ Thus we have four $G$-spaces of $\phi$-scalars $(\mathbb{R},G,h_{\phi })$ in the geometry of the group $G.$ The group $G$ operates also on the sphere $S^{n-2}$ forming a $G$-space of isotropic directions $(S^{n-2},G,\ast ).$ In this note, we have solved the functional equation $F(A\ast q_1,A\ast q_2,\dots ,A\ast q_m)=\phi (A)\cdot F(q_1,q_2,\dots ,q_m)$ for given independent points $q_1,q_2,\dots ,q_m\in S^{n-2}$ with $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby\ we have determined all equivariant mappings $F:(S^{n-2}{)}^m\to \mathbb{R}.$

Détail

DOI : 10.21136/MB.2008.140618

Classification : 53A55
Mots-clés : $G$-space; equivariant map; pseudo-Euclidean geometry