Geodesic
Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$
Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 265-275

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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}.$
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}.$
DOI : 10.21136/MB.2005.134097
Classification : 22E99, 53A35, 53A55
Keywords: $G$-space; equivariant map; pseudo-Euclidean geometry; functional equation 
Glanc, Barbara; Misiak, Aleksander; Stepień, Zofia. Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$. Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 265-275. doi: 10.21136/MB.2005.134097
@article{10_21136_MB_2005_134097,
     author = {Glanc, Barbara and Misiak, Aleksander and Stepie\'n, Zofia},
     title = {Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$},
     journal = {Mathematica Bohemica},
     pages = {265--275},
     year = {2005},
     volume = {130},
     number = {3},
     doi = {10.21136/MB.2005.134097},
     mrnumber = {2164656},
     zbl = {1108.53009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134097/}
}
TY  - JOUR
AU  - Glanc, Barbara
AU  - Misiak, Aleksander
AU  - Stepień, Zofia
TI  - Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$
JO  - Mathematica Bohemica
PY  - 2005
SP  - 265
EP  - 275
VL  - 130
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134097/
DO  - 10.21136/MB.2005.134097
LA  - en
ID  - 10_21136_MB_2005_134097
ER  - 
%0 Journal Article
%A Glanc, Barbara
%A Misiak, Aleksander
%A Stepień, Zofia
%T Equivariant mappings from vector product into $G$-space of vectors and $\varepsilon $-vectors with $G=O(n,1,\mathbb{R})$
%J Mathematica Bohemica
%D 2005
%P 265-275
%V 130
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2005.134097/
%R 10.21136/MB.2005.134097
%G en
%F 10_21136_MB_2005_134097

[1] J. Aczél, S. Gołąb: Functionalgleichungen der Theorie der geometrischen Objekte. P.W.N. Warszawa, 1960. | MR

[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

[3] M. Kucharzewski: Über die Grundlagen der Kleinschen Geometrie. Period. Math. Hung. 8 (1977), 83–89. | DOI | MR | Zbl

[4] A. Misiak, E. Stasiak: Equivariant maps between certain $G$-spaces with $G=O( n-1,n)$. Math. Bohem. 126 (2001), 555–560. | MR

[5] E. Stasiak: Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1. Publ. Math. Debrecen 57 (2000), 55–69. | MR | Zbl

Cité par Sources :