Lie derivative



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids




\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



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

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,

v=[d dR,ι v]=d dRι v+ι vd 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) \,.


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


Last revised on April 4, 2021 at 00:21:16. See the history of this page for a list of all contributions to it.