Editing Euclidean space (section)

==Other geometric spaces==
{{split section|Geometric space|discuss=Talk:Geometric_space#Splitting_from_Euclidean_n-space|date=March 2023}}

Since the introduction, at the end of 19th century, of [[non-Euclidean geometries]], many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical [[axiom]]s, [[embedding]] the space in a Euclidean space is a standard way for proving [[consistency]] of its definition, or, more precisely for proving that its theory is consistent, if [[Euclidean geometry]] is consistent (which cannot be proved).

===Affine space===
{{main|Affine space}}
A Euclidean space is an affine space equipped with a [[metric (mathematics)|metric]]. Affine spaces have many other uses in mathematics. In particular, as they are defined over any [[field (mathematics)|field]], they allow doing geometry in other contexts.

As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the [[complex number]]s as an extension of Euclidean spaces. For example, a [[circle]] and a [[line (geometry)|line]] have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of [[algebraic geometry]] is built in complex affine spaces and affine spaces over [[algebraically closed field]]s. The shapes that are studied in algebraic geometry in these affine spaces are therefore called [[affine algebraic variety|affine algebraic varieties]].

Affine spaces over the [[rational number]]s and more generally over [[algebraic number field]]s provide a link between (algebraic) geometry and [[number theory]]. For example, the [[Fermat's Last Theorem]] can be stated "a [[Fermat curve]] of degree higher than two has no point in the affine plane over the rationals."

Geometry in affine spaces over a [[finite fields]] has also been widely studied. For example, [[elliptic curve]]s over finite fields are widely used in [[cryptography]].

===Projective space===
{{main|Projective space}}
Originally, projective spaces have been introduced by adding "[[points at infinity]]" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two [[coplanar]] lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being [[isotropic]], that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the [[vector line]]s in a [[vector space]] of dimension one more.

As for affine spaces, projective spaces are defined over any [[field (mathematics)|field]], and are fundamental spaces of [[algebraic geometry]].

===Non-Euclidean geometries===
{{main|Non-Euclidean geometry}}
''Non-Euclidean geometry'' refers usually to geometrical spaces where the [[parallel postulate]] is false. They include [[elliptic geometry]], where the sum of the angles of a triangle is more than 180°, and [[hyperbolic geometry]], where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is [[consistency|consistent]] (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the [[foundational crisis in mathematics]] of the beginning of 20th century, and motivated the systematization of [[axiomatic theory|axiomatic theories]] in mathematics.

=== Curved spaces ===
{{main|Curved space}}
{{further|Manifold|Riemannian manifold}}

A [[manifold]] is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a [[topological space]], such that each point has a [[neighborhood]] that is [[homeomorphic]] to an [[open subset]] of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into [[topological manifold]]s, [[differentiable manifold]]s, [[smooth manifold]]s, and [[analytic manifold]]s. However, none of these types of "resemblance" respect distances and angles, even approximately.

Distances and angles can be defined on a smooth manifold by providing a [[smooth function|smoothly varying]] Euclidean metric on the [[tangent space]]s at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a [[Riemannian manifold]]. Generally, [[straight line]]s do not exist in a Riemannian manifold, but their role is played by [[geodesic]]s, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent.

Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a [[sphere]]. In this case, geodesics are [[great circle|arcs of great circle]], which are called [[orthodrome]]s in the context of [[navigation]]. More generally, the spaces of [[non-Euclidean geometries]] can be realized as Riemannian manifolds.

=== Pseudo-Euclidean space ===
An [[inner product]] of a real vector space is a [[positive definite bilinear form]], and so characterized by a [[Bilinear form#Derived quadratic form|positive definite quadratic form]]. A [[pseudo-Euclidean space]] is an affine space with an associated real vector space equipped with a [[non-degenerate]] [[quadratic form]] (that may be [[indefinite quadratic form|indefinite]]).

A fundamental example of such a space is the [[Minkowski space]], which is the [[space-time]] of [[Albert Einstein|Einstein]]'s [[special relativity]]. It is a four-dimensional space, where the metric is defined by the [[quadratic form]]

<math display="block">x^2+y^2+z^2-t^2,</math>

where the last coordinate (''t'') is temporal, and the other three (''x'', ''y'', ''z'') are spatial.

To take [[gravity]] into account, [[general relativity]] uses a [[pseudo-Riemannian manifold]] that has Minkowski spaces as [[tangent space]]s. The [[Curvature of Riemannian manifolds|curvature]] of this manifold at a point is a function of the value of the [[gravitational field]] at this point.