# 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.

#### Comments

The Riemannian curvature is better known as the sectional curvature.

For references see Riemann tensor.

**How to Cite This Entry:**

Riemannian curvature.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Riemannian_curvature&oldid=48558