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

Misiak, Aleksander, and Stasiak, Eugeniusz. "$G$-space of isotropic directions and $G$-spaces of $ \varphi $-scalars with $G=O( n,1,\mathbb {R}) $." Mathematica Bohemica 133.3 (2008): 289-298. <http://eudml.org/doc/250540>.

@article{Misiak2008,
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 $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F\colon ( S^\{n-2\}) ^m\rightarrow \mathbb \{R\}.$},
author = {Misiak, Aleksander, Stasiak, Eugeniusz},
journal = {Mathematica Bohemica},
keywords = {$G$-space; equivariant map; pseudo-Euclidean geometry; -space; equivariant map; pseudo-Euclidean geometry},
language = {eng},
number = {3},
pages = {289-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$G$-space of isotropic directions and $G$-spaces of $ \varphi $-scalars with $G=O( n,1,\mathbb \{R\}) $},
url = {http://eudml.org/doc/250540},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Misiak, Aleksander
AU - Stasiak, Eugeniusz
TI - $G$-space of isotropic directions and $G$-spaces of $ \varphi $-scalars with $G=O( n,1,\mathbb {R}) $
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 289
EP - 298
AB - 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 $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F\colon ( S^{n-2}) ^m\rightarrow \mathbb {R}.$
LA - eng
KW - $G$-space; equivariant map; pseudo-Euclidean geometry; -space; equivariant map; pseudo-Euclidean geometry
UR - http://eudml.org/doc/250540
ER -