# Pseudo-Riemannian space

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A space with an affine connection (without torsion), at each point of which the tangent space is a pseudo-Euclidean space.

Let be an -space with an affine connection (without torsion) and let be the tangent pseudo-Euclidean space at every point of ; in this case the pseudo-Riemannian space is denoted by . As in a proper Riemannian space the metric tensor of is non-degenerate, has vanishing covariant derivative, but the metric form of is a quadratic form of index :

is the metric tensor of , . The space can be defined as an -dimensional manifold on which an invariant quadratic differential form of index is given.

The simplest example of a pseudo-Riemannian space is the space .

The pseudo-Riemannian space is said to be reducible if in a neighbourhood of each point there is a system of coordinates such that the coordinates can all be separated into groups such that only for those indices and which belong to a single group and the are functions only of the coordinates of this group.

In a pseudo-Riemannian space the sectional curvature is defined for every non-degenerate two-dimensional direction. It can be interpreted as the curvature of the geodesic (non-isotropic) -surface drawn through the given point in the given two-dimensional direction. If the value of the curvature at each point is the same for all two-dimensional directions, then it is constant at all points (Schur's theorem), and in this case the space is said to be a pseudo-Riemannian space of constant curvature . An example of a pseudo-Riemannian space of constant negative curvature is the hyperbolic space of negative curvature — it is a pseudo-Riemannian space ; the space is a pseudo-Riemannian space of vanishing curvature.

#### References

 [1] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) [2] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) [3] A. Einstein, "Collected scientific works" , 1 , Moscow (1965) (In Russian; translated from English)