# Semi-pseudo-Euclidean space

A vector space with a degenerate indefinite metric. The semi-pseudo-Euclidean space is defined as an -dimensional space in which there are given scalar products

where ; ; ; , and occurs times among the numbers . The product is defined for those vectors for which all coordinates , or , are zero. The first scalar square of an arbitrary vector of a semi-pseudo-Euclidean space is a degenerate quadratic form in the vector coordinates:

where is the index and is the defect of the space. If , the semi-pseudo-Euclidean space is a semi-Euclidean space. Straight lines, -dimensional planes , parallelism, and length of vectors, are defined in semi-pseudo-Euclidean spaces in the same way as in pseudo-Euclidean spaces. In the semi-pseudo-Euclidean space one can choose an orthogonal basis consisting of vectors of imaginary length, of of real length and of isotropic vectors. Through every point of a semi-pseudo-Euclidean space of defect passes an -dimensional isotropic plane all vectors of which are orthogonal to all vectors of the space. See also Galilean space.

#### References

[1] | 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-pseudo-Euclidean space. D.D. Sokolov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Semi-pseudo-Euclidean_space&oldid=11931