Previous |  Up |  Next

Article

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
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 $\mathbb{R}_{0}=\left( \mathbb{R}\setminus \left\rbrace 0\right\lbrace ,\cdot \right)$.
Keyword: $G$-space
Keyword: equivariant map
Keyword: pseudo-Euclidean geometry
MSC: 53A55
idZBL: Zbl 1174.53007
idMR: MR2355661
.
Date available: 2009-09-24T22:32:10Z
Last updated: 2012-06-18
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
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
.

Files

Files Size Format View
MathBohem_132-2007-3_7.pdf 224.0Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo