nLab Lie derivative

Idea

Given a smooth manifold $M$ and a vector field $X \in \Gamma(T M)$ on it, one defines a series of operators $\mathcal{L}_X$ on spaces of differential forms, of functions, of vector fields and multivector fields. For functions $\mathcal{L}_X(f) = X(f)$ (derivative of $f$ along an integral curve of $X$); 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 $\mathcal{L}_X Y = [X,Y]$. If $\omega \in \Omega^\bullet(M)$ is a differential form on $M$, the Lie derivative $\mathcal{L}_X \omega$ of $\omega$ along $X$ is the linearization of the pullback of $\omega$ along the flow $\exp(X -) : \mathbb{R} \times M\to M$ induced by $X$

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

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

$$\mathcal{L}_v = [d_{dR}, \iota_v] = d_{dR} \circ \iota_v + \iota_v \circ d_{dR} : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.$$

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

On covariant Lie derivatives of bundle-valued differential forms:

• Grigorios Giotopoulos: Covariant Lie Derivatives and (Super-)Gravity [arXiv:2507.00140]