View source for Quasi-Euclidean space

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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,   "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>