# Riemannian curvature

A measure of the difference between the metrics of a Riemannian and a Euclidean space. Let $M$ be a point of a Riemannian space and let $F$ be a two-dimensional regular surface $x ^ {i} = x ^ {i} ( u, v)$ passing through $M$, let $L$ be a simply closed contour in $F$ passing through $M$, and let $\sigma$ be the area of the part of $F$ bounded by $L$. Apply the parallel displacement along $L$ to an arbitrary vector $a ^ {i}$ tangent to $F$( that is, a linear expression in the vectors $\partial x ^ {i} / \partial u$, $\partial x ^ {i} / \partial v$). Then the component of the transferred vector tangential to $F$ turns out to be turned in relation to $a ^ {i}$ by an angle $\phi$( the positive reference direction of the angle must coincide with the direction of movement along $L$). If, when $L$ is contracted to the point $M$, the limit

$$K = \lim\limits \frac \phi \sigma$$

exists, then it is called the Riemannian curvature (the 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 $M$, that is, on the direction of the two-dimensional tangent plane to the Euclidean space that contains the vectors $\partial ^ {i} x/ \partial u$, $\partial x ^ {i} / \partial v$.

The Riemannian curvature $K$ is connected with the curvature tensor by the formula:

$$K = \sum _ {m,l,k,j } R _ {mlkj} x ^ {ml} x ^ {kj} ,$$

where

$$x ^ {ml} = \frac{1}{2} \left ( \frac{\partial x ^ {m} }{\partial u } \frac{\partial x ^ {l} }{\partial v } - \frac{\partial x ^ {l} }{\partial u } \frac{\partial x ^ {m} }{\partial v } \right ) ,$$

in which the parameters $u , v$ are chosen such that the area of the parallelogram constructed on the vectors $\partial x ^ {i} / \partial u$, $\partial x ^ {i} / \partial v$ equals 1.