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.