In this note all vectors and ε-vectors of a system of m ≤ n linearly independent contravariant vectors in the n-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation F(Au 1 , Au 2 , . . . , Au m ) = (det A) λ · A · F(u 1 , u 2 , . . . , u m ) with λ = 0 and λ = 1, for an arbitrary pseudo-orthogonal matrix A of index one and given vectors u 1 , u 2 , . . . , u m .
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 ≤ 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(Au 1 , Au 2 , . . . , Au m ) = ϕ (A) · F(u 1 , u 2 , . . . , u m ) using two homomorphisms ϕ from a group G into the group of real numbers R0 = (R \ {0} , ·).
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.