Difference between revisions of "Riemannian curvature"
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| − | + | A measure of the difference between the metrics of a Riemannian and a Euclidean space. Let $ M $ | |
| + | be a point of a [[Riemannian space|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(2)|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|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|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==== | ====Comments==== | ||
Latest revision as of 08:11, 6 June 2020
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.
Riemannian curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_curvature&oldid=19233