# Semi-Euclidean space

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.

#### 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) |

#### Comments

#### 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. L.A. Sidorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Semi-Euclidean_space&oldid=12128