Quasi-Euclidean space

A $2$-dimensional space in which each direction given at a point of it can be included in a field the directions of which can be transferred parallelly along any path (that is, a quasi-Euclidean space admits an absolute parallelism). The geodesic lines of a quasi-Euclidean space are partitioned into $\infty^1$ families of vector lines of fields of absolutely parallel directions, where each such family forms with three others a constant cross ratio:

$$\frac{k-k_1}{k_2-k}:\frac{k_3-k_1}{k_2-k_3}=\text{const},$$

where $k=du^2/du^1$ is the angular direction coefficient. Each family of geodesics is defined in terms of three constants by a first-order equation:

$$\frac{a_pdu^p}{b_qdu^q}=\text{const}.$$

References

[1]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

References

[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)