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Difference between revisions of "Curvature transformation"

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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273701.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273702.png" /> of vector fields on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273703.png" />, depending linearly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273704.png" /> and given by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273705.png" /></td> </tr></table>
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here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273706.png" /> is the [[Covariant derivative|covariant derivative]] in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273708.png" /> is the Lie bracket of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c0273709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c02737010.png" />. The mapping
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A mapping  $  R ( X, Y) $
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of the space  $  {\mathcal T} ( M) $
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of vector fields on a manifold  $  M $,
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depending linearly on  $  X, Y \in {\mathcal T} ( M) $
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and given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c02737011.png" /></td> </tr></table>
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$$
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R ( X, Y) Z  = \
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\nabla _ {X} \nabla _ {Y} Z -
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\nabla _ {Y} \nabla _ {X} Z -
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\nabla _ {[ X, Y] }  Z;
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$$
  
is the [[Curvature tensor|curvature tensor]] of the [[Linear connection|linear connection]] defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027370/c02737012.png" />.
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here  $  \nabla _ {X} $
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is the [[Covariant derivative|covariant derivative]] in the direction of  $  X $
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and  $  [ X, Y] $
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is the Lie bracket of  $  X $
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and  $  Y $.
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The mapping
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$$
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R  \equiv \
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R ( X, Y) Z:  {\mathcal T}  ^ {3} ( M)  \rightarrow  {\mathcal T} ( M)
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$$
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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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article