Lie derivative in nLab

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Contents

Idea

Given a smooth manifold MM and a vector field X∈Γ(TM)X \in \Gamma(T M) on it, one defines a series of operators ℒ X\mathcal{L}_X on spaces of differential forms, of functions, of vector fields and multivector fields. For functions ℒ X(f)=X(f)\mathcal{L}_X(f) = X(f) (derivative of ff along an integral curve of XX); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.

For vector fields ℒ XY=[X,Y]\mathcal{L}_X Y = [X,Y]. If ω∈Ω •(M)\omega \in \Omega^\bullet(M) is a differential form on MM, the Lie derivative ℒ Xω\mathcal{L}_X \omega of ω\omega along XX is the linearization of the pullback of ω\omega along the flow exp(X−):ℝ×M→M\exp(X -) : \mathbb{R} \times M\to M induced by XX

ℒ Xω=ddt| t=0exp(tX) *(ω). \mathcal{L}_X \omega = \frac{d}{d t}|_{t = 0} \exp(t X)^*(\omega) \,.

Denote by ι X:Ω •(M)→Ω •−1(M)\iota_X : \Omega^\bullet(M) \to \Omega^{\bullet -1}(M) be the graded derivation which is the contraction with a vector field XX. By Cartan's homotopy formula,

ℒ v=[d dR,ι v]=d dR∘ι v+ι v∘d dR:Ω •(X)→Ω •(X). \mathcal{L}_v = [d_{dR}, \iota_v] = d_{dR} \circ \iota_v + \iota_v \circ d_{dR} : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

References

Cartan introduced Lie derivatives of differential forms and derived Cartan's magic formula in

• Élie Cartan, Leçons sur les invariants intégraux (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

Extension to arbitrary tensor fields was given in

• W. Ślebodziński, Sur les équations de Hamilton, Bull. Acad. Roy. de Belg. 17 (1931).

The term “Lie derivative” (Liesche Ableitung) is due to van Dantzig, who also suggested a definition using the flow of a vector field:

• D. van Dantzig, Zur allgemeinen projektiven Differentialgeometrie, Proc. Roy. Acad. Amsterdam 35 (1932) Part I: 524–534; Part II: 535–542.

An introduction in the context of synthetic differential geometry is in

• Gonzalo Reyes, Lie derivatives, Lie brackets and vector fields over curves, pdf

A gentle elementary introduction for mathematical physicists

• Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro) amazon, google

.