In this note, there are determined all biscalars of a system of s ≤ 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(Au1 , Au2 , . . . , Aus ) = (sign(det A))F(u1 , u2 ,...,us ) for an arbitrary pseudo-orthogonal matrix A of index one and the given vectors u1 , u2 ,...,us .
There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O(n, 1, R) into the group of real numbers R0. Thus we have four G-spaces of ϕ-scalars (R, G, hϕ) 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∗q1, A∗q2, . . . , A∗qm) = ϕ(A)·F(q1, q2, . . . , qm) for given independent points q1, q2, . . . , qm ∈ S n−2 with 1 ≤ m ≤ n and an arbitrary matrix A ∈ G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : (S n−2 ) m → R.