View source for Covariant derivative

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:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026850/c0268505.png" />,

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:

<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>

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)]]).

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]].



====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.