# 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.

#### 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) Zbl 0767.53035 Zbl 0713.53012 Zbl 0702.53009 [3] A. Einstein, "Collected scientific works" , 1 , Moscow (1965) (In Russian; translated from English)

Important for applications in physics is the case of index 1. For any $n$, ${} ^ {1} V _ {n}$ is called a Lorentzian manifold. Here the tangent vectors having purely imaginary, vanishing or non-vanishing real length are called time-like, light-like or space-like, respectively. If $n = 4$ and ${} ^ {1} V _ {n}$ possesses a global time-like vector field, then ${} ^ {1} V _ {n}$ is called a space-time and admits a distinction between future and past. Such a space-time is the general geometric model in which the phenomena of the theory of general relativity are described. For example, the history of a material or light-like particle is described by a world-line having non-space-like tangents only. The coupling between the geometric model and the physical data is given by Einstein's field equations (cf. Einstein equations).