Barbara Glanc, Aleksander Misiak, Maria Szmuksta-Zawadzka, Instytut Matematyki, Politechnika Szczecinska, Al. Piastow 17, 70-310 Szczecin, Poland, e-mail: misiak@ps.pl
Abstract: 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 $m\leq n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F( A\underset1\to{u},A \underset2\to{u},\dots,A\underset{m}\to{u}) = \varphi\left( A\right) \cdot F( \underset1\to{u},\underset2\to{u},\dots,\underset{m}\to{u})$ using two homomorphisms $\varphi$ from a group $G$ into the group of real numbers $\mathbb{R}_0=\left( \mathbb{R}\setminus\left\{ 0\right\} ,\cdot\right)$.
Keywords: $G$-space, equivariant map, pseudo-Euclidean geometry
Classification (MSC 2000): 53A55
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