# Pseudo-Riemannian space

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

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

$$d s ^ {2} = g _ {ij} d x ^ {i} d x ^ {j} ,\ \ i , j = 1 \dots n ,$$

$g _ {ij}$ is the metric tensor of ${} ^ {l} V _ {n}$, $\mathop{\rm det} \| g _ {ij} \| \neq 0$. The space ${} ^ {l} V _ {n}$ can be defined as an $n$- dimensional manifold on which an invariant quadratic differential form of index $l$ is given.

The simplest example of a pseudo-Riemannian space is the space ${} ^ {l} \mathbf R _ {n}$.

The pseudo-Riemannian space ${} ^ {l} V _ {n}$ is said to be reducible if in a neighbourhood of each point there is a system of coordinates $( x ^ {1} \dots x ^ {n} )$ such that the coordinates $x ^ {i}$ can all be separated into groups $x ^ {i _ \alpha }$ such that $g _ {i _ \alpha j _ \alpha } \neq 0$ only for those indices $i _ \alpha$ and $j _ \alpha$ which belong to a single group and the $g _ {i _ \alpha j _ \alpha }$ 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) $2$- 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 $( n \geq 3 )$. An example of a pseudo-Riemannian space of constant negative curvature is the hyperbolic space ${} ^ {l} S _ {n}$ of negative curvature — it is a pseudo-Riemannian space ${} ^ {n-} l V _ {n}$; the space ${} ^ {l} \mathbf R _ {n}$ is a pseudo-Riemannian space of vanishing curvature.

How to Cite This Entry:
Pseudo-Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-Riemannian_space&oldid=48343
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article