View source for Riemannian curvature

A measure of the difference between the metrics of a Riemannian and a Euclidean space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821301.png" /> be a point of a [[Riemannian space|Riemannian space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821302.png" /> be a two-dimensional regular surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821303.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821304.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821305.png" /> be a simply closed contour in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821306.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821307.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821308.png" /> be the area of the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r0821309.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213010.png" />. Apply the [[Parallel displacement(2)|parallel displacement]] along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213011.png" /> to an arbitrary vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213012.png" /> tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213013.png" /> (that is, a linear expression in the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213015.png" />). Then the component of the transferred vector tangential to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213016.png" /> turns out to be turned in relation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213017.png" /> by an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213018.png" /> (the positive reference direction of the angle must coincide with the direction of movement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213019.png" />). If, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213020.png" /> is contracted to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213021.png" />, the limit

<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/r/r082/r082130/r08213022.png" /></td> </tr></table>

exists, then it is called the Riemannian curvature (the [[Curvature|curvature]] of the Riemannian space) at the given point in the direction of the two-dimensional surface; the Riemannian curvature does not depend on the surface but only on its direction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213023.png" />, that is, on the direction of the two-dimensional tangent plane to the Euclidean space that contains the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213025.png" />.

The Riemannian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213026.png" /> is connected with the [[Curvature tensor|curvature tensor]] 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/r/r082/r082130/r08213027.png" /></td> </tr></table>

where

<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/r/r082/r082130/r08213028.png" /></td> </tr></table>

in which the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213029.png" /> are chosen such that the area of the parallelogram constructed on the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082130/r08213031.png" /> equals 1.



====Comments====
The Riemannian curvature is better known as the sectional curvature.

For references see [[Riemann tensor|Riemann tensor]].