Covariant Derivative - Relation To Lie Derivative

Relation To Lie Derivative

A covariant derivative introduces an extra geometric structure on a manifold which allows vectors in neighboring tangent spaces to be compared. This extra structure is necessary because there is no canonical way to compare vectors from different vector spaces, as is necessary for this generalization of the directional derivative. There is however another generalization of directional derivatives which is canonical: the Lie derivative. The Lie derivative evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood. 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 an open neighborhood of a point. In other words, the covariant derivative is linear (over C∞(M)) in the direction argument, while the Lie derivative is linear in neither argument.

Note that the antisymmetrized covariant derivative ∇_uv − ∇_vu, and the Lie derivative L_uv differ by the torsion of the connection, so that if a connection is symmetric, then its antisymmetrization is the Lie derivative.