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Semi-pseudo-Riemannian space

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A manifold with a degenerate indefinite metric. The semi-pseudo-Riemannian manifold is defined as an -dimensional manifold with coordinates in which there are given line elements

where ; ; and where the index of the quadratic form is . The line element is defined for those vectors for which all components with indices smaller than or larger than vanish. If , a semi-pseudo-Riemannian space is a semi-Riemannian space. The spaces and are quasi-Riemannian spaces. The basic concepts of differential geometry (for example, curvature) are defined in semi-pseudo-Riemannian spaces similarly to Riemannian spaces (see [1]).

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-Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-pseudo-Riemannian_space&oldid=48665
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article