| Title:
|
$G$-space of isotropic directions and $G$-spaces of $ \varphi $-scalars with $G=O( n,1,\mathbb R) $ (English) |
| Author:
|
Misiak, Aleksander |
| Author:
|
Stasiak, Eugeniusz |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
133 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
289-298 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $F\colon ( S^{n-2}) ^m\rightarrow \mathbb R.$ (English) |
| Keyword:
|
$G$-space |
| Keyword:
|
equivariant map |
| Keyword:
|
pseudo-Euclidean geometry |
| MSC:
|
53A55 |
| idZBL:
|
Zbl 1199.53034 |
| idMR:
|
MR2494782 |
| DOI:
|
10.21136/MB.2008.140618 |
| . |
| Date available:
|
2010-07-20T17:30:11Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140618 |
| . |
| Reference:
|
[1] Aczél, J., Gołąb, S.: Functionalgleichungen der Theorie der geometrischen Objekte.P.W.N. Warszawa (1960). |
| Reference:
|
[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. MR 0971951 |
| Reference:
|
[3] Kucharzewski, M.: Über die Grundlagen der Kleinschen Geometrie.Period. Math. Hungar. 8 (1977), 83-89. Zbl 0335.50001, MR 0493695, 10.1007/BF02018051 |
| Reference:
|
[4] Misiak, A., Stasiak, E.: Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$.Math. Bohem. 126 (2001), 555-560. Zbl 1031.53031, MR 1970258 |
| Reference:
|
[5] Stasiak, E.: On a certain action of the pseudoorthogonal group with index one $O( n,1,\mathbb R)$ on the sphere $S^{n-2}$.Polish Prace Naukowe P.S. 485 (1993). |
| Reference:
|
[6] Stasiak, E.: Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1.Publ. Math. Debrecen 57 (2000), 55-69. Zbl 0966.53012, MR 1771671 |
| . |
