View source for Non-Euclidean space

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A space whose properties are based on a system of axioms other than the Euclidean system. The geometries of non-Euclidean spaces are the [[Non-Euclidean geometries|non-Euclidean geometries]]. Depending on the specific axioms from which the non-Euclidean geometries are developed in non-Euclidean spaces, the latter may be classified in accordance with various criteria. On the one hand, a non-Euclidean space may be a finite-dimensional vector space with a scalar product expressible in Cartesian coordinates as

$$ 
( \mathbf a , \mathbf b )  = \ 
\sum _ {i = 1 } ^ { k }  x _ {i} y _ {i} -
\sum _ {i = k + 1 } ^ { n }  x _ {i} y _ {i} .
$$

In this case one speaks of a [[Pseudo-Euclidean space|pseudo-Euclidean space]]. On the other hand, a non-Euclidean space can be characterized as an  $  n $-
dimensional manifold with a certain structure described by a non-Euclidean axiom system.

Non-Euclidean spaces may also be classified from the point of view of their differential-geometric properties as Riemannian spaces of constant curvature (this includes the case of spaces of curvature zero, which are nevertheless topologically distinct from Euclidean spaces).

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Greenberg,   "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Rosenfeld,   "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>