Lie derivative (for vector fields)

Let M be a smooth manifoldMathworldPlanetmath, and X,Y𝒯(M) smooth vector fields on M. Let Θ:𝒰M be the flow of X, where 𝒰×M is an open neighborhood of {0}×M. We make use of the following notation:


and we introduce the auxiliary maps θt:𝒰tM and θp:𝒰pM defined as


The Lie derivativeMathworldPlanetmathPlanetmath of Y along X is the vector field XY𝒯(M) defined by


where d(θ-t)θt(p)Hom(Tθt(p)M,TpM) if the push-forward of θ-t, i.e.


The following result is not immediate at all.

Theorem 1

XY=[X,Y], where [X,Y]=XY-YX is the Lie bracket of X and Y.

Title Lie derivative (for vector fields)
Canonical name LieDerivativeforVectorFields
Date of creation 2013-03-22 14:09:59
Last modified on 2013-03-22 14:09:59
Owner matte (1858)
Last modified by matte (1858)
Numerical id 9
Author matte (1858)
Entry type Definition
Classification msc 53-00
Defines Lie derivative