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The Weyl curvature tensor for the generalized Reissner-Nordstrοm space-time is determined and theorems related to the Penrose conjecture are proved.

In this note all vectors and $\epsilon$-vectors of a system of $m\le n$ 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 $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)={(detA)}^{\lambda }·A·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ with $\lambda =0$ and $\lambda =1$, for an arbitrary pseudo-orthogonal matrix $A$ of index one and given vectors $\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u.$

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 $m\le n$ linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation $F(A\underset{1}{\to }u,A\underset{2}{\to }u,\cdots ,A\underset{m}{\to }u)=\varphi \left(A\right)·F(\underset{1}{\to }u,\underset{2}{\to }u,\cdots ,\underset{m}{\to }u)$ using two homomorphisms $\varphi$ from a group $G$ into the group of real numbers ${ℝ}_0=\left(ℝ\setminus \left\}0\right\{,·\right)$.