Differential geometry MOC

Lie derivative

Let be a vector field generating the flow . The Lie derivative

is an -linear map defined by¹ diff

It is also possible to define the Lie derivative along a vector at a single point.

Algebraic properties

1. Agreement with other derivatives: .
2. Leibniz rule: .
3. Commutes with contraction: .

Relation to the covariant derivative

For any affine connexion , we have

Proof

proof

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develop | en | sembr

Footnotes

1. 2009. General relativity, pp. 439–441, C.2 ↩