Editing Covariant derivative (section)

==Relation to Lie derivative==
A covariant derivative introduces an extra geometric structure on a manifold that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.

There is however another generalization of directional derivatives which ''is'' canonical: the [[Lie derivative]], which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in a neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in a neighborhood of a point. In other words, the covariant derivative is linear (over {{math|''C''{{isup|∞}}(''M'')}}) in the direction argument, while the Lie derivative is linear in neither argument.

Note that the antisymmetrized covariant derivative {{math|∇<sub>''u''</sub>''v'' − ∇<sub>''v''</sub>''u''}}, and the Lie derivative {{math|''L''<sub>''u''</sub>''v''}} differ by the [[torsion of connection|torsion of the connection]], so that if a connection is torsion free, then its antisymmetrization ''is'' the Lie derivative.