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Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.
Euclidean spaces are also Riemann spaces. Smooth surfaces in Euclidean spaces are Riemann spaces. A hyperbolic non-Euclidean space is also a Riemann space. A curve in a Riemann space has a length, and the length of the shortest curve between two points defines a distance, such that the Riemann space is a metric space.
In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat, as in the Euclidean space. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. [3]
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). A fundamental example of such a space is the Minkowski space , which is the space-time of Einstein 's special relativity .
Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (1, 3) or (3, 1). Elements of Minkowski space are called events. Minkowski space is often denoted R 1,3 or R 3,1 to emphasize the chosen signature, or just M. It is an example of a pseudo-Riemannian ...
The points of n-dimensional projective space can be identified with lines through the origin in (n + 1)-dimensional space, and can be represented non-uniquely by nonzero vectors in R n+1, with the understanding that u and λu, for any non-zero scalar λ, represent the same point. Distance is defined using the metric
The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.
In mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate quadratic form q.Such a quadratic form can, given a suitable choice of basis (e 1, …, e n), be applied to a vector x = x 1 e 1 + ⋯ + x n e n, giving = (+ +) (+ + +) which is called the scalar square of the vector x.