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 $ \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.