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$G$-space; equivariant map; pseudo-Euclidean geometry
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 $F\colon ( S^{n-2}) ^m\rightarrow \mathbb R.$
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