# Equivariant maps between certain $G$-spaces with $G=O\left(n-1,1\right)$.

Mathematica Bohemica (2001)

• Volume: 126, Issue: 3, page 555-560
• ISSN: 0862-7959

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## Abstract

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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 $F\left(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{s}{\to }u\right)=\left(\text{sign}\left(detA\right)\right)F\left(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u\right)$ for an arbitrary pseudo-orthogonal matrix $A$ of index one and the given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{s}{\to }u$.

## How to cite

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Misiak, Aleksander, and Stasiak, Eugeniusz. "Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$.." Mathematica Bohemica 126.3 (2001): 555-560. <http://eudml.org/doc/248881>.

@article{Misiak2001,
abstract = {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 $\{\underset\{1\}\{\rightarrow \}u\}, \{\underset\{2\}\{\rightarrow \}u\},\dots ,\{\underset\{s\}\{\rightarrow \}u\}$.},
author = {Misiak, Aleksander, Stasiak, Eugeniusz},
journal = {Mathematica Bohemica},
keywords = {$G$-space; equivariant map; vector; scalar; biscalar; -space; equivariant map; vector; scalar; biscalar},
language = {eng},
number = {3},
pages = {555-560},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$.},
url = {http://eudml.org/doc/248881},
volume = {126},
year = {2001},
}

TY - JOUR
AU - Misiak, Aleksander
AU - Stasiak, Eugeniusz
TI - Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$.
JO - Mathematica Bohemica
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 3
SP - 555
EP - 560
AB - 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 ${\underset{1}{\rightarrow }u}, {\underset{2}{\rightarrow }u},\dots ,{\underset{s}{\rightarrow }u}$.
LA - eng
KW - $G$-space; equivariant map; vector; scalar; biscalar; -space; equivariant map; vector; scalar; biscalar
UR - http://eudml.org/doc/248881
ER -

## References

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1. Funktionalgleichungen der Theorie der geometrischen Objekte, P.W.N Warszawa, 1960. (1960) MR0133763
2. Sur deux formes équivalentes de la notion de $\left(r,s\right)$-orientation de la géométrie de Klein, Publ. Math. Debrecen 35 (1988), 43–50. (1988) MR0971951
3. Invariant Theory, Academic Press, New York, 1971. (1971) MR0279102
4. Über die Grundlagen der Kleinschen Geometrie, Period. Math. Hung. 8 (1977), 83–89. (1977) Zbl0335.50001MR0493695
5. O pewnym działaniu grupy pseudoortogonalnej o indeksie jeden $O\left(n,1,R\right)$ na sferze ${S}^{n-2}$, Prace Naukowe P. S., 485, Szczecin, 1993. (1993)
6. Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1, Publ. Math. Debrecen 57 (2000), 55–69. (2000) Zbl0966.53012MR1771671

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