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

Abstract: There exist exactly four homomorphisms $\varphi$ 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 $\varphi$-scalars $( \mathbb{R},G,h_{\varphi}) $ 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}) =\varphi( 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 $1\leq m\leq n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F ( S^{n-2}) ^{m}\rightarrow\mathbb{R}.$

Keywords: $G$-space, equivariant map, pseudo-Euclidean geometry

Classification (MSC2000): 53A55

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