Difference between revisions of "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 | 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 | + | 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). | 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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====Comments==== | ====Comments==== | ||
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====References==== | ====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> | <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> | ||
Latest revision as of 08:02, 6 June 2020
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).
Comments
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) |
Non-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-Euclidean_space&oldid=16086