Difference between revisions of "Covariant differential"
From Encyclopedia of Mathematics
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| − | A generalization of the notion of a differential to fields of different geometric objects. It is a tensor | + | 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 | + | 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=32828
Covariant differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_differential&oldid=32828
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article