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There exist exactly four homomorphisms $\varphi$ from the pseudo-orthogonal group of index one $G=O(n,1,ℝ)$ into the group of real numbers ${ℝ}_0.$ Thus we have four $G$-spaces of $\varphi$-scalars $(ℝ,G,h_{\varphi })$ 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 $(S^{n-2},G,*).$ In this note, we have solved the functional equation $F(A*q_1,A*q_2,\cdots ,A*q_m)=\varphi \left(A\right)·F(q_1,q_2,\cdots ,q_m)$ for given independent points $q_1,q_2,\cdots ,q_m\in S^{n-2}$ with $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F:{\left(S^{n-2}\right)}^m\to ℝ.$ ...

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