# Semi-Euclidean space

A real affine $n$- space equipped with a scalar product of vectors such that, relative to a suitably chosen basis, the scalar product $( \mathbf x , \mathbf x )$ of any vector with itself has the form

$$( \mathbf x , \mathbf x ) = - \sum _ { i= } 1 ^ { l } ( x ^ {i} ) ^ {2} + \sum _ { j= } l+ 1 ^ { n- } d ( x ^ {j} ) ^ {2} .$$

Under these conditions, the semi-Euclidean space is said to have index $l$ and deficiency $d$ and is denoted by ${} ^ {l + ( d ) } R _ {n}$. If $l = 0$, the expression for the scalar product of a vector with itself is a semi-definite quadratic form and the space is called an $n$- space of deficiency $d$, denoted by ${} ^ {( d ) } R _ {n}$.

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 $n$- space which is a hypersphere with identified antipodal points in the semi-Euclidean space of index $l$ and deficiency $d$. 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 ${} ^ {( 1 ) } R _ {n}$( see [2]).

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

#### References

 [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)

#### References

 [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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-Euclidean_space&oldid=48651
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article