# Semi-pseudo-Euclidean space

A vector space with a degenerate indefinite metric. The semi-pseudo-Euclidean space ${} ^ {l _ {1} \dots l _ {r} } R _ {n} ^ {m _ {1} \dots m _ {r - 1 } }$ is defined as an $n$- dimensional space in which there are given $r$ scalar products

$$( x, y) _ {a} = \ \sum \epsilon _ {i _ {a} } x ^ {i _ {a} } y ^ {i _ {a} } ,$$

where $0 = m _ {0} < m _ {1} < \dots < m _ {r} = n$; $a = 1 \dots r$; $i _ {a} = m _ {a - 1 } + 1 \dots m _ {a}$; $\epsilon _ {i _ {a} } = \pm 1$, and $- 1$ occurs $l _ {a}$ times among the numbers $\epsilon _ {i _ {a} }$. The product $( x, y) _ {a}$ is defined for those vectors for which all coordinates $x ^ {i}$, $i \leq m _ {a - 1 }$ or $i> m _ {a} + 1$, are zero. The first scalar square of an arbitrary vector $x$ of a semi-pseudo-Euclidean space is a degenerate quadratic form in the vector coordinates:

$$( x, x) = \ - ( x _ {1} ) ^ {2} - \dots - ( x _ {l _ {1} } ) ^ {2} + ( x _ {l _ {1} + 1 } ) ^ {2} + \dots + ( x _ {n - d } ) ^ {2} ,$$

where $l$ is the index and $d = n - m _ {1}$ is the defect of the space. If $l _ {1} = \dots = l _ {r} = 0$, the semi-pseudo-Euclidean space is a semi-Euclidean space. Straight lines, $m$- dimensional planes $( m < n)$, 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 ${} ^ {l + ( d) } {R _ {n} }$ one can choose an orthogonal basis consisting of $l$ vectors of imaginary length, of $n - l - d$ of real length and of $d$ isotropic vectors. Through every point of a semi-pseudo-Euclidean space of defect $d$ passes an $n$- dimensional isotropic plane all vectors of which are orthogonal to all vectors of the space. See also Galilean space.

How to Cite This Entry:
Semi-pseudo-Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-pseudo-Euclidean_space&oldid=48664
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article