Pseudo-Riemannian geometry

The totality of geometric properties of surfaces and curves in a pseudo-Riemannian space $ {} ^ {l} V _ {n} $. These properties arise from the properties of a pseudo-Riemannian metric on this space, which is an indefinite quadratic form of index $ l $:

$$ d s ^ {2} = g _ {ij} ( X) d x ^ {i} d x ^ {j} . $$

The arc length of a curve $ l $ is expressed by the formula

$$ s = \int\limits _ { l } \sqrt {g _ {ij} d x ^ {i} d x ^ {j} } ; $$

it may be real, purely imaginary or zero (an isotropic curve). A geodesic curve in $ {} ^ {l} V _ {n} $ loses the property of being, even locally, extremal, but remains a curve of stationary length. The length of an arc $ l $ can be greater or less than the length of the geodesic segment joining the ends of $ l $. If one considers the space $ {} ^ {n- 1} V _ {n} $, then a geodesic segment $ A B $ of real length gives the longest distance between the points $ A $ and $ B $ (assuming that its arc can be imbedded in a semi-geodesic coordinate system as a coordinate line and that for comparison smooth curves of real length are taken from the domain in which this coordinate system is defined). If the pseudo-Riemannian space $ {} ^ {n-} 1 \mathbf R _ {n} $ is being considered, then any straight line of real length can be taken as the $ x ^ {n} $-axis of an orthogonal coordinate system in which the scalar square of a vector $ \mathbf x $ has the form

$$ \langle \mathbf x , \mathbf x \rangle = - \sum _ { i= 1} ^ { n- 1} ( x ^ {i} ) ^ {2} + ( x ^ {n} ) ^ {2} . $$

Here any straight line segment of real length (along the $ x ^ {n} $-axis) gives the longest distance between its end-points. In the case of the space $ {} ^ {1} V _ {n} $ (or $ {} ^ {1} \mathbf R _ {n} $) a segment of a geodesic line with imaginary length will give the longest distance in comparison with all possible smooth curves with imaginary length the ends of which coincide with the ends of the geodesic segment.

Using a pseudo-Riemannian metric one can develop the differential geometry of surfaces and curves in a pseudo-Riemannian space, define the curvature of curves and surfaces, etc.

Pseudo-Riemannian geometry also arises on surfaces in hyperbolic spaces. The simplest case of a pseudo-Riemannian geometry is the geometry of a pseudo-Euclidean space $ {} ^ {l} \mathbf R _ {n} $ and, in particular, the geometry of Minkowski space.

For references and more information see Pseudo-Riemannian space.