G -space of isotropic directions and G -spaces of ϕ -scalars with G = O ( n , 1 , )

Aleksander Misiak; Eugeniusz Stasiak

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 289-298
  • ISSN: 0862-7959

Abstract

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There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O ( n , 1 , ) into the group of real numbers 0 . Thus we have four G -spaces of ϕ -scalars ( , G , h ϕ ) 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 , * ) . In this note, we have solved the functional equation F ( A * q 1 , A * q 2 , , A * q m ) = ϕ ( A ) · F ( q 1 , q 2 , , q m ) for given independent points q 1 , q 2 , , q m S n - 2 with 1 m n and an arbitrary matrix A G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : ( S n - 2 ) m .

How to cite

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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 -

References

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  1. Aczél, J., Gołąb, S., Functionalgleichungen der Theorie der geometrischen Objekte, P.W.N. Warszawa (1960). (1960) 
  2. Bieszk, L., Stasiak, E., Sur deux formes équivalents de la notion de ( r , s ) -orientation de la géometrié de Klein, Publ. Math. Debrecen 35 (1988), 43-50. (1988) MR0971951
  3. Kucharzewski, M., Über die Grundlagen der Kleinschen Geometrie, Period. Math. Hungar. 8 (1977), 83-89. (1977) Zbl0335.50001MR0493695
  4. Misiak, A., Stasiak, E., Equivariant maps between certain G -spaces with G = O ( n - 1 , 1 ) , Math. Bohem. 126 (2001), 555-560. (2001) Zbl1031.53031MR1970258
  5. Stasiak, E., On a certain action of the pseudoorthogonal group with index one O ( n , 1 , ) on the sphere S n - 2 , Polish Prace Naukowe P.S. 485 (1993). (1993) 
  6. Stasiak, E., Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1, Publ. Math. Debrecen 57 (2000), 55-69. (2000) Zbl0966.53012MR1771671

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