Difference between revisions of "Quasi-Euclidean space"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| − | A | + | {{TEX|done}} |
| + | 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 | + | 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==== | ====References==== | ||
Latest revision as of 15:03, 1 May 2014
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) |
Comments
References
| [a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
How to Cite This Entry:
Quasi-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-Euclidean_space&oldid=17183
Quasi-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-Euclidean_space&oldid=17183
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article