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Non-Euclidean space

From Encyclopedia of Mathematics
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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. 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).

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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)
How to Cite This Entry:
Non-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-Euclidean_space&oldid=16086
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article