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 $F:{\left(S^{n-2}\right)}^m\to ℝ.$

We show how the ad hoc prescriptions appearing in 2001 for the Lie derivative of Lorentz tensors are a direct consequence of the Kosmann lift defined earlier, in a much more general setting encompassing older results of Y. Kosmann about Lie derivatives of spinors.

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups $\mathrm{GA}\left(2\right)=\mathrm{GL}(2,ℝ)⋉{ℝ}^2$ and $\mathrm{GA}\left(3\right)=\mathrm{GL}(3,ℝ)⋉{ℝ}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective...