| Title:
|
Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$. (English) |
| Author:
|
Misiak, Aleksander |
| Author:
|
Stasiak, Eugeniusz |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
555-560 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note, there are determined all biscalars of a system of $s\le n$ linearly independent contravariant vectors in $n$-dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation $F(A{\underset{1}{\rightarrow }u},A {\underset{2}{\rightarrow }u},\dots ,A{\underset{s}{\rightarrow }u}) =( \text{sign}( \det A)) F ({\underset{1}{\rightarrow }u},{\underset{2}{\rightarrow }u},\dots ,{\underset{s}{\rightarrow }u}) $ for an arbitrary pseudo-orthogonal matrix $A$ of index one and the given vectors ${\underset{1}{\rightarrow }u}, {\underset{2}{\rightarrow }u},\dots ,{\underset{s}{\rightarrow }u}$. (English) |
| Keyword:
|
$G$-space |
| Keyword:
|
equivariant map |
| Keyword:
|
vector |
| Keyword:
|
scalar |
| Keyword:
|
biscalar |
| MSC:
|
53A55 |
| idZBL:
|
Zbl 1031.53031 |
| idMR:
|
MR1970258 |
| DOI:
|
10.21136/MB.2001.134200 |
| . |
| Date available:
|
2009-09-24T21:54:07Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134200 |
| . |
| Reference:
|
[1] J. Aczél, S. Gołb: Funktionalgleichungen 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] J. A. Dieudonné, J. B. Carrell: Invariant Theory.Academic Press, New York, 1971. MR 0279102 |
| Reference:
|
[4] M. Kucharzewski: Über die Grundlagen der Kleinschen Geometrie.Period. Math. Hung. 8 (1977), 83–89. Zbl 0335.50001, MR 0493695, 10.1007/BF02018051 |
| Reference:
|
[5] E. Stasiak: O pewnym działaniu grupy pseudoortogonalnej o indeksie jeden $O(n,1,R)$ na sferze $S^{n-2}$.Prace Naukowe P. S., 485, Szczecin, 1993. |
| Reference:
|
[6] 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 |
| . |
