Lie Derivative - Properties

Properties

The Lie derivative has a number of properties. Let be the algebra of functions defined on the manifold M. Then

is a derivation on the algebra . That is, is R-linear and

Similarly, it is a derivation on where is the set of vector fields on M:

which may also be written in the equivalent notation

$\mathcal{L}_X(f\otimes Y)= (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y$

where the tensor product symbol is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.

Additional properties are consistent with that of the Lie bracket. Thus, for example, considered as a derivation on a vector field,

one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then

$\alpha:= \mathcal{L}_X\mathcal{L}_Y\alpha-\mathcal{L}_Y\mathcal{L}_X\alpha=\mathcal{L}_{}\alpha$
where i denotes interior product defined above and it's clear whether denotes the commutator or the Lie bracket of vector fields.

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