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A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. It is a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268501.png" /> acting on the module of tensor fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268502.png" /> of given valency and defined with respect to a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268503.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268504.png" /> and satisfying the following properties:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268505.png" />,
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2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268506.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268508.png" /> are differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268509.png" />. This mapping is trivially extended by linearity to the algebra of tensor fields and one additionally requires for the action on tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685010.png" /> of different valency:
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A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. It is a linear operator  $  \nabla _ {X} $
 +
acting on the module of tensor fields $  T _ {s} ^ { r } ( M) $
 +
of given valency and defined with respect to a vector field  $  X $
 +
on a manifold  $  M $
 +
and satisfying the following properties:
  
<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/c026850/c02685011.png" /></td> </tr></table>
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1)  $  \nabla _ {f X + g Y }  U = f \nabla _ {X} U + g \nabla _ {Y} U $,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685012.png" /> denotes the tensor product. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685013.png" /> is a derivation on the algebra of tensor fields (cf. [[Derivation in a ring|Derivation in a ring]]); it has the additional properties of commuting with operations of contraction (cf. [[Contraction of a tensor|Contraction of a tensor]]), skew-symmetrization (cf. [[Alternation|Alternation]]) and symmetrization of tensors (cf. [[Symmetrization (of tensors)|Symmetrization (of tensors)]]).
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2)  $  \nabla _ {X} ( f U ) = f \nabla _ {X} U + ( X f ) U $,  
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where  $  U \in T _ {s} ^ { r } ( M) $
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and $  f , g $
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are differentiable functions on  $  M $.  
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This mapping is trivially extended by linearity to the algebra of tensor fields and one additionally requires for the action on tensors $  U , V $
 +
of different valency:
  
Properties 1) and 2) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685014.png" /> (for vector fields) allow one to introduce on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685015.png" /> a linear connection (and the corresponding parallel displacement) and on the basis of this, to give a local definition of a covariant derivative which, when extended to the whole manifold, coincides with the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c02685016.png" /> defined above; see also [[Covariant differentiation|Covariant differentiation]].
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$$
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\nabla _ {X} ( U \otimes V ) = \
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\nabla _ {X} U \otimes V + U
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\otimes \nabla _ {X} V ,
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$$
  
 +
where  $  \otimes $
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denotes the tensor product. Thus  $  \nabla _ {X} $
 +
is a derivation on the algebra of tensor fields (cf. [[Derivation in a ring|Derivation in a ring]]); it has the additional properties of commuting with operations of contraction (cf. [[Contraction of a tensor|Contraction of a tensor]]), skew-symmetrization (cf. [[Alternation|Alternation]]) and symmetrization of tensors (cf. [[Symmetrization (of tensors)|Symmetrization (of tensors)]]).
  
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Properties 1) and 2) of  $  \nabla _ {X} $(
 +
for vector fields) allow one to introduce on  $  M $
 +
a linear connection (and the corresponding parallel displacement) and on the basis of this, to give a local definition of a covariant derivative which, when extended to the whole manifold, coincides with the operator  $  \nabla _ {X} $
 +
defined above; see also [[Covariant differentiation|Covariant differentiation]].
  
 
====Comments====
 
====Comments====
 
There is not much of a difference between the notions of a covariant derivative and covariant differentiation and both are used in the same context.
 
There is not much of a difference between the notions of a covariant derivative and covariant differentiation and both are used in the same context.

Latest revision as of 17:31, 5 June 2020


A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. It is a linear operator $ \nabla _ {X} $ acting on the module of tensor fields $ T _ {s} ^ { r } ( M) $ of given valency and defined with respect to a vector field $ X $ on a manifold $ M $ and satisfying the following properties:

1) $ \nabla _ {f X + g Y } U = f \nabla _ {X} U + g \nabla _ {Y} U $,

2) $ \nabla _ {X} ( f U ) = f \nabla _ {X} U + ( X f ) U $, where $ U \in T _ {s} ^ { r } ( M) $ and $ f , g $ are differentiable functions on $ M $. This mapping is trivially extended by linearity to the algebra of tensor fields and one additionally requires for the action on tensors $ U , V $ of different valency:

$$ \nabla _ {X} ( U \otimes V ) = \ \nabla _ {X} U \otimes V + U \otimes \nabla _ {X} V , $$

where $ \otimes $ denotes the tensor product. Thus $ \nabla _ {X} $ is a derivation on the algebra of tensor fields (cf. Derivation in a ring); it has the additional properties of commuting with operations of contraction (cf. Contraction of a tensor), skew-symmetrization (cf. Alternation) and symmetrization of tensors (cf. Symmetrization (of tensors)).

Properties 1) and 2) of $ \nabla _ {X} $( for vector fields) allow one to introduce on $ M $ a linear connection (and the corresponding parallel displacement) and on the basis of this, to give a local definition of a covariant derivative which, when extended to the whole manifold, coincides with the operator $ \nabla _ {X} $ defined above; see also Covariant differentiation.

Comments

There is not much of a difference between the notions of a covariant derivative and covariant differentiation and both are used in the same context.

How to Cite This Entry:
Covariant derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_derivative&oldid=46543
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article