View source for Pseudo-Riemannian geometry

The totality of geometric properties of surfaces and curves in a [[Pseudo-Riemannian space|pseudo-Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758001.png" />. These properties arise from the properties of a pseudo-Riemannian metric on this space, which is an indefinite quadratic form of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758002.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758003.png" /></td> </tr></table>

The arc length of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758004.png" /> is expressed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758005.png" /></td> </tr></table>

it may be real, purely imaginary or zero (an isotropic curve). A geodesic curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758006.png" /> loses the property of being, even locally, extremal, but remains a curve of stationary length. The length of an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758007.png" /> can be greater or less than the length of the geodesic segment joining the ends of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758008.png" />. If one considers the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p0758009.png" />, then a geodesic segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580010.png" /> of real length gives the longest distance between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580012.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580013.png" /> is being considered, then any straight line of real length can be taken as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580014.png" />-axis of an orthogonal coordinate system in which the scalar square of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580015.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580016.png" /></td> </tr></table>

Here any straight line segment of real length (along the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580017.png" />-axis) gives the longest distance between its end-points. In the case of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580018.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580019.png" />) 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|pseudo-Euclidean space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075800/p07580020.png" /> and, in particular, the geometry of [[Minkowski space|Minkowski space]].

For references and more information see [[Pseudo-Riemannian space|Pseudo-Riemannian space]].