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\(G\)-space of isotropic directions and \(G\)-spaces of \(\varphi \)-scalars with \(G=O(n,1,\mathbb {R})\). (English) Zbl 1199.53034
Summary: 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 \(F\:(S^{n-2})^{m}\rightarrow \mathbb {R}.\)

53A55 Differential invariants (local theory), geometric objects
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