Lie commutator

In differential geometry, the Lie derivative /ˈliː/, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow of another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted ${\displaystyle {\mathcal {L}}_{X}(T)}$. The differential operator ${\displaystyle T\mapsto {\mathcal {L}}_{X}(T)}$ is a derivation of the algebra of tensor fields of the underlying manifold.

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