Non-Euclidean space

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. 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. 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).

References

[a1]	M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a2]	B. Rosenfeld, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)