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Covariant derivative

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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 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