| Title:
|
Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$ (English) |
| Author:
|
Glanc, Barbara |
| Author:
|
Misiak, Aleksander |
| Author:
|
Stepień, Zofia |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
130 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
265-275 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note all vectors and $\varepsilon $-vectors of a system of $m\le n$ linearly independent contravariant vectors in the $n$-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation $F( A{\underset{1}{\rightarrow }u}, A{\underset{2}{\rightarrow }u},\dots ,A{\underset{m}{\rightarrow }u}) =( \det A)^{\lambda }\cdot A\cdot F( {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots , {\underset{m}{\rightarrow }u})$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $ {\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots ,{\underset{m}{\rightarrow }u}.$ (English) |
| Keyword:
|
$G$-space |
| Keyword:
|
equivariant map |
| Keyword:
|
pseudo-Euclidean geometry |
| Keyword:
|
functional equation |
| MSC:
|
22E99 |
| MSC:
|
53A35 |
| MSC:
|
53A55 |
| idZBL:
|
Zbl 1108.53009 |
| idMR:
|
MR2164656 |
| DOI:
|
10.21136/MB.2005.134097 |
| . |
| Date available:
|
2009-09-24T22:20:59Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134097 |
| . |
| Reference:
|
[1] J. Aczél, S. Gołąb: Functionalgleichungen der Theorie der geometrischen Objekte.P.W.N. Warszawa, 1960. MR 0133763 |
| Reference:
|
[2] L. Bieszk, E. Stasiak: Sur deux formes équivalentes de la notion de $( r,s)$-orientation de la géométrie de Klein.Publ. Math. Debrecen 35 (1988), 43–50. MR 0971951 |
| Reference:
|
[3] M. Kucharzewski: Über die Grundlagen der Kleinschen Geometrie.Period. Math. Hung. 8 (1977), 83–89. Zbl 0335.50001, MR 0493695, 10.1007/BF02018051 |
| Reference:
|
[4] A. Misiak, E. Stasiak: Equivariant maps between certain $G$-spaces with $G=O( n-1,n)$.Math. Bohem. 126 (2001), 555–560. MR 1970258 |
| Reference:
|
[5] 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 |
| . |
