# 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.