# Article

Full entry | PDF   (0.2 MB)
Keywords:
$G$-space; equivariant map; pseudo-Euclidean geometry
Summary:
There exist exactly four homomorphisms $\varphi$ from the pseudo-orthogonal group of index one $G=O( n,1,\mathbb R)$ into the group of real numbers $\mathbb R_0.$ Thus we have four $G$-spaces of $\varphi$-scalars $( \mathbb R,G,h_{\varphi })$ in the geometry of the group $G.$ The group $G$ operates also on the sphere $S^{n-2}$ forming a $G$-space of isotropic directions $( S^{n-2},G,\ast ) .$ In this note, we have solved the functional equation $F( A\ast q_1,A\ast q_2,\dots ,A\ast q_m) =\varphi ( A) \cdot F( q_1,q_2,\dots ,q_m)$ for given independent points $q_1,q_2,\dots ,q_m\in S^{n-2}$ with $1\leq m\leq n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby\ we have determined all equivariant mappings $F\colon ( S^{n-2}) ^m\rightarrow \mathbb R.$
References:
[1] Aczél, J., Gołąb, S.: Functionalgleichungen der Theorie der geometrischen Objekte. P.W.N. Warszawa (1960).
[2] Bieszk, L., Stasiak, E.: Sur deux formes équivalents de la notion de $(r,s)$-orientation de la géometrié de Klein. Publ. Math. Debrecen 35 (1988), 43-50. MR 0971951
[3] Kucharzewski, M.: Über die Grundlagen der Kleinschen Geometrie. Period. Math. Hungar. 8 (1977), 83-89. DOI 10.1007/BF02018051 | MR 0493695 | Zbl 0335.50001
[4] Misiak, A., Stasiak, E.: Equivariant maps between certain $G$-spaces with $G=O( n-1,1)$. Math. Bohem. 126 (2001), 555-560. MR 1970258 | Zbl 1031.53031
[5] Stasiak, E.: On a certain action of the pseudoorthogonal group with index one $O( n,1,\mathbb R)$ on the sphere $S^{n-2}$. Polish Prace Naukowe P.S. 485 (1993).
[6] Stasiak, E.: Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1. Publ. Math. Debrecen 57 (2000), 55-69. MR 1771671 | Zbl 0966.53012

Partner of