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.

References

[1]	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)