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Difference between revisions of "Covariant differential"

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A generalization of the notion of a differential to fields of different geometric objects. It is a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268601.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268602.png" /> on a manifold with values in the module of tensor fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268603.png" /> defined by the formula
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A generalization of the notion of a differential to fields of different geometric objects. It is a tensor $1$-form $DU$ on a manifold with values in the module of tensor fields $U$ defined 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/c026/c026860/c0268604.png" /></td> </tr></table>
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$$(DU)(X)=\nabla_XU,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268605.png" /> is the [[Covariant derivative|covariant derivative]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268606.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026860/c0268607.png" />. For detailed information, see [[Covariant differentiation|Covariant differentiation]].
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where $\nabla_XU$ is the [[Covariant derivative|covariant derivative]] of the field $U$ along $X$. For detailed information, see [[Covariant differentiation|Covariant differentiation]].

Latest revision as of 15:33, 10 August 2014

A generalization of the notion of a differential to fields of different geometric objects. It is a tensor $1$-form $DU$ on a manifold with values in the module of tensor fields $U$ defined by the formula

$$(DU)(X)=\nabla_XU,$$

where $\nabla_XU$ is the covariant derivative of the field $U$ along $X$. For detailed information, see Covariant differentiation.

How to Cite This Entry:
Covariant differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_differential&oldid=32829
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article