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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 $$F\:(S^{n-2})^{m}\rightarrow \mathbb {R}.$$

##### MSC:
 53A55 Differential invariants (local theory), geometric objects
##### Keywords:
$$G$$-space; equivariant map; pseudo-Euclidean geometry
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