Semi-Euclidean space

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A real affine -space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product of any vector with itself has the form

Under these conditions, the semi-Euclidean space is said to have index and deficiency and is denoted by . If , the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an -space of deficiency , denoted by .

In the projective classification, a semi-Euclidean space can be defined as a semi-elliptic space or a semi-hyperbolic space with an improper absolute plane; these are spaces with projective metrics of the most general form.

One defines a semi-non-Euclidean space as a metric -space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index and deficiency . Thus, semi-elliptic and semi-hyperbolic spaces may be interpreted as hyperspheres of the above type (i.e. as semi-non-Euclidean spaces) in semi-Euclidean spaces of appropriate index and deficiency.

The geometrical interpretation of Galileo–Newton mechanics leads to the semi-Euclidean space (see [2]).

A semi-Euclidean space is a semi-Riemannian space of curvature zero.


[1] D.M.Y. Sommerville, "Classification of geometries with projective metric" Proc. Edinburgh Math. Soc. , 28 (1910) pp. 25–41
[2] A.P. Kotel'nikov, "The principle of relativity and Lobachevskii geometry" , In memoriam N.I. Lobachevskii , 2 , Kazan' (1926) (In Russian)
[3] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)



[a1] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
How to Cite This Entry:
Semi-Euclidean space. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098