| Title:
|
Equivariant mappings from vector product into $ G$-spaces of $\varphi $-scalars with $G=O\left( n,1,\mathbb{R}\right) $ (English) |
| Author:
|
Glanc, Barbara |
| Author:
|
Misiak, Aleksander |
| Author:
|
Szmuksta-Zawadzka, Maria |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
132 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
325-332 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
There are four kinds of scalars in the $n$-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F( A\underset{1}{\rightarrow }{u},A \underset{2}{\rightarrow }{u},\dots ,A\underset{m}{\rightarrow }{u}) = \varphi \left( A\right) \cdot F( \underset{1}{\rightarrow }{u},\underset{2}{\rightarrow }{u},\dots ,\underset{m}{\rightarrow }{u})$ using two homomorphisms $\varphi $ from a group $G$ into the group of real numbers $\mathbb{R}_{0}=\left( \mathbb{R}\setminus \left\rbrace 0\right\lbrace ,\cdot \right)$. (English) |
| Keyword:
|
$G$-space |
| Keyword:
|
equivariant map |
| Keyword:
|
pseudo-Euclidean geometry |
| MSC:
|
53A55 |
| idZBL:
|
Zbl 1174.53007 |
| idMR:
|
MR2355661 |
| DOI:
|
10.21136/MB.2007.134120 |
| . |
| Date available:
|
2009-09-24T22:32:10Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134120 |
| . |
| Reference:
|
[1] J. Aczél, S. Gołąb: Functionalgleichungen der Theorie der geometrischen Objekte.Panstwowe Wydawnietvo Naukove, Warszawa, 1960. MR 0133763 |
| Reference:
|
[2] L. Bieszk, E. Stasiak: Sur deux formes équivalents de la notion de $(r,s)$-orientation de la géométrie de Klein.Publ. Math. Debrecen 35 (1988), 43–50. MR 0971951 |
| Reference:
|
[3] E. Kasparek: The homomorphisms of the pseudo-orthogonal group of index one into an abelian group.Demonstratio Math. 22 (1989), 763–771. MR 1041913 |
| Reference:
|
[4] M. Kucharzewski: Über die Grundlagen der Kleinschen Geometrie.Period. Math. Hungar. 8 (1977), 83–89. Zbl 0335.50001, MR 0493695, 10.1007/BF02018051 |
| Reference:
|
[5] A. Misiak, E. Stasiak: Equivariant maps between certain $G$-spaces with $G=O\left( n-1,1\right)$.Math. Bohem. 126 (2001), 555–560. MR 1970258 |
| Reference:
|
[6] E. Stasiak: O pewnym działaniu grupy pseudoortogonalnej o indeksie jeden $O\left( n,1,\mathbb{R}\right) $ na sferze $S^{n-2}$.Prace Naukowe P.S. 485 (1993). |
| Reference:
|
[7] E. Stasiak: 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 |
| . |
