Difference between revisions of "Curvature transformation"
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| − | + | A mapping $ R ( X, Y) $ | |
| + | of the space $ {\mathcal T} ( M) $ | ||
| + | of vector fields on a manifold $ M $, | ||
| + | depending linearly on $ X, Y \in {\mathcal T} ( M) $ | ||
| + | and given by the formula | ||
| − | + | $$ | |
| + | R ( X, Y) Z = \ | ||
| + | \nabla _ {X} \nabla _ {Y} Z - | ||
| + | \nabla _ {Y} \nabla _ {X} Z - | ||
| + | \nabla _ {[ X, Y] } Z; | ||
| + | $$ | ||
| − | is the [[Curvature tensor|curvature tensor]] of the [[Linear connection|linear connection]] defined by | + | here $ \nabla _ {X} $ |
| + | is the [[Covariant derivative|covariant derivative]] in the direction of $ X $ | ||
| + | and $ [ X, Y] $ | ||
| + | is the Lie bracket of $ X $ | ||
| + | and $ Y $. | ||
| + | The mapping | ||
| + | |||
| + | $$ | ||
| + | R \equiv \ | ||
| + | R ( X, Y) Z: {\mathcal T} ^ {3} ( M) \rightarrow {\mathcal T} ( M) | ||
| + | $$ | ||
| + | |||
| + | is the [[Curvature tensor|curvature tensor]] of the [[Linear connection|linear connection]] defined by $ \nabla _ {X} $. | ||
Latest revision as of 17:31, 5 June 2020
A mapping $ R ( X, Y) $
of the space $ {\mathcal T} ( M) $
of vector fields on a manifold $ M $,
depending linearly on $ X, Y \in {\mathcal T} ( M) $
and given by the formula
$$ R ( X, Y) Z = \ \nabla _ {X} \nabla _ {Y} Z - \nabla _ {Y} \nabla _ {X} Z - \nabla _ {[ X, Y] } Z; $$
here $ \nabla _ {X} $ is the covariant derivative in the direction of $ X $ and $ [ X, Y] $ is the Lie bracket of $ X $ and $ Y $. The mapping
$$ R \equiv \ R ( X, Y) Z: {\mathcal T} ^ {3} ( M) \rightarrow {\mathcal T} ( M) $$
is the curvature tensor of the linear connection defined by $ \nabla _ {X} $.
How to Cite This Entry:
Curvature transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_transformation&oldid=46566
Curvature transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_transformation&oldid=46566
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article