The Lie Derivative

Introduction

The Lie derivative along a vector field $X$ differentiates the tensors of a manifold by transporting them along the flow of $X$ and comparing the transported object with the original. It needs no metric and no connection, and it is therefore the derivative that the smooth structure alone supplies; where a connection is present the covariant derivative and the Lie derivative are different operators, and their difference is a tensor built from the derivative of the field.

The article develops the Lie derivative as an operator. It defines it by the flow on functions, on vector fields and on forms, proves that it is a derivation of degree zero of the graded algebra of forms commuting with the exterior derivative, and develops the interior product and Cartan's formula $\mathcal{L}_X = d\iota_X + \iota_X d$. It derives the identities $[\mathcal{L}_X, \mathcal{L}_Y] = \mathcal{L}_{[X,Y]}$, $[\mathcal{L}_X, \iota_Y] = \iota_{[X,Y]}$ and $\mathcal{L}_X d = d\mathcal{L}_X$, computes the Lie derivative of a volume form as the divergence, and relates $\mathcal{L}_X$ to the covariant derivative of a torsion-free connection by the formula in which the tensor $\nabla X$ appears.

The article assumes the smooth manifolds, the vector fields, their Lie bracket and the differential of a smooth map of Smooth Manifolds and Differential Geometry; the differential forms, the wedge product and the exterior derivative of Differential Forms and Stokes' Theorem; and the covariant derivative of a connection of The Covariant Derivative in this category. The flow of a vector field is the local one-parameter group of its differential equation, whose existence and uniqueness are Ordinary Differential Equations, later in this Part, and are quoted where used. The Killing fields, the isometries and the Lie derivative of a metric read as geometry are Part IV's, and are named as forward references. No physics is invoked.

The Lie Derivative of Functions and Vector Fields

The Flow of a Vector Field

Definition. Let $X \in \mathrm{X}(M)$ be a vector field on a smooth manifold $M$. A local flow of $X$ is a smooth map $\phi : (-\epsilon, \epsilon) \times U \to M$ defined on an open set $U$ such that

$$ \phi(0, x) = x, \qquad \frac{d}{dt}\phi(t, x) = X_{\phi(t,x)} $$

for all $(t,x)$ in the domain; the maps $\phi_t = \phi(t, \cdot)$ are the local diffeomorphisms generated by $X$, and they satisfy the group law $\phi_s \circ \phi_t = \phi_{s+t}$ wherever both sides are defined.

Theorem. About every point there is a neighbourhood and a time interval on which the local flow of $X$ exists and is unique.

Proof. The equation $\dot\gamma = X(\gamma)$ is a first-order ordinary differential equation with a smooth vector field on the right-hand side; the theorem of Ordinary Differential Equations, later in this Part, gives a unique solution through each initial condition. Uniqueness gives the group law: both $t\mapsto\phi_{s+t}(x)$ and $t\mapsto\phi_s(\phi_t(x))$ solve the equation with the same value at $t=0$. No global flow is asserted: a vector field on a non-compact manifold may have solutions that escape in finite time, and the flow is local.

Functions and Vector Fields

Definition. Let $f \in C^\infty(M)$ and $X \in \mathrm{X}(M)$. The Lie derivative of $f$ along $X$ is the function

$$ \mathcal{L}_X f = \frac{d}{dt}\Big|_{t=0} \phi_t^* f = X(f) = df(X), $$

the directional derivative of $f$ along $X$.

Definition. The Lie derivative of a vector field $Y$ along $X$ is the vector field

$$ \mathcal{L}_X Y = \frac{d}{dt}\Big|_{t=0} (\phi_{-t})_* Y = [X, Y], $$

the bracket of Smooth Manifolds and Differential Geometry; the second equality is the definition of the bracket by the flow, and the two earlier definitions of $[X,Y]$ — as the derivation $X(Yf) - Y(Xf)$ and as the derivative of the pushed field — agree.

Proposition. The assignment $Y \mapsto \mathcal{L}_X Y$ is $\mathbb{R}$-linear, satisfies the Leibniz rule $\mathcal{L}_X(fY) = (Xf)Y + f\mathcal{L}_XY$, and makes $\mathrm{X}(M)$ a Lie algebra with $[\mathcal{L}_X, \mathcal{L}_Y] = \mathcal{L}_{[X,Y]}$.

Proof. These are the standard properties of the bracket, proved in Smooth Manifolds and Differential Geometry for the derivation form; the flow form gives the same identities because the derivative of the conjugation by $\phi_t$ is a derivation of the algebra of fields under the bracket, and the Jacobi identity is the associativity of the composition of the three flows.

The Lie Derivative of Forms and Tensor Fields

Extension by the Flow

Definition. Let $\omega \in \Omega^k(M)$. The Lie derivative of $\omega$ along $X$ is the form

$$ \mathcal{L}_X\omega = \frac{d}{dt}\Big|_{t=0} \phi_t^*\omega, $$

the derivative of the pullback of $\omega$ along the flow.

Proposition. The Lie derivative $\mathcal{L}_X$ is the unique degree-zero derivation of the graded algebra $\Omega^\bullet(M)$ with

$$ \mathcal{L}_X f = Xf \quad \text{on functions}, \qquad \mathcal{L}_X\alpha = d\iota_X\alpha + \iota_X d\alpha \quad \text{on forms}, $$

and the two formulas agree; on a general tensor field it is defined the same way by the flow, and it is the unique derivation of the tensor algebra that commutes with the contractions and extends $\mathcal{L}_Xf = Xf$ and $\mathcal{L}_XY = [X,Y]$.

Proof. The derivative of the pullback satisfies $\frac{d}{dt}|_0\phi_t^*(\omega\wedge\eta) = \mathcal{L}_X\omega\wedge\eta + \omega\wedge\mathcal{L}_X\eta$, because $\phi_t^*$ is an algebra homomorphism and the product rule differentiates the product; so $\mathcal{L}_X$ is a derivation. It preserves the degree because the pullback does. On functions it is $Xf$ by the first definition; the extension by the derivation property is forced, and it is realised by the flow. On a general $(p,q)$-tensor the two rules $\mathcal{L}_X(T\otimes S) = \mathcal{L}_XT\otimes S + T\otimes\mathcal{L}_XS$ and the commutation with the contractions determine the action, and the flow realises it.

Proposition. The Lie derivative commutes with the exterior derivative,

$$ \mathcal{L}_X(d\omega) = d(\mathcal{L}_X\omega), $$

and with the pullback along every diffeomorphism: $F^*(\mathcal{L}_XF^*\omega) = \mathcal{L}_{F^*X}(F^*\omega)$ for a diffeomorphism $F$.

Proof. The exterior derivative commutes with the pullback along any smooth map, $F^* d = d F^*$, so $\mathcal{L}_X d\omega = \frac{d}{dt}|_0\phi_t^* d\omega = \frac{d}{dt}|_0 d\phi_t^*\omega = d\mathcal{L}_X\omega$. The second statement is the naturality of the flow: the flow of $F^*X$ is $F^{-1}\circ\phi_t\circ F$.

The Interior Product and Cartan's Formula

The Interior Product

Definition. Let $X \in \mathrm{X}(M)$. The interior product (or contraction) with $X$ is the $\mathbb{R}$-linear operator $\iota_X : \Omega^k(M) \to \Omega^{k-1}(M)$ defined by

$$ (\iota_X\omega)(Y_1, \ldots, Y_{k-1}) = \omega(X, Y_1, \ldots, Y_{k-1}), $$

with $\iota_X = 0$ on $\Omega^0(M)$.

Proposition. The interior product is a degree $-1$ derivation of the graded algebra of forms, $\iota_X(\alpha\wedge\beta) = \iota_X\alpha\wedge\beta + (-1)^p\alpha\wedge\iota_X\beta$ for $\alpha \in \Omega^p(M)$; it is $C^\infty(M)$-linear in $X$, $\iota_{fX} = f\iota_X$; it satisfies $\iota_X^2 = 0$ and $\iota_X\iota_Y + \iota_Y\iota_X = 0$; and on a $1$-form it is the evaluation, $\iota_X\alpha = \alpha(X)$.

Proof. The first identity is the antisymmetry of the alternating forms: evaluating on $X$ and inserting it into the first slot, the sign of the move past $\alpha$ of degree $p$ is $(-1)^p$. The square vanishes because two contractions in the same slot of an alternating form cancel; the module linearity is immediate from the definition; the evaluation statement is the definition for $k=1$.

Cartan's Formula

Theorem (Cartan's formula). For every vector field $X$ and every form $\omega$,

$$ \mathcal{L}_X\omega = d\,\iota_X\omega + \iota_X\, d\omega, $$

that is, $\mathcal{L}_X = d\,\iota_X + \iota_X\,d$ as operators on $\Omega^\bullet(M)$.

Proof. Both sides are degree-zero derivations of the graded algebra of forms: the right-hand side is the sum of the compositions of the derivations $d$ and $\iota_X$, which is a derivation, and the left-hand side is a derivation by the previous section. Two derivations of $\Omega^\bullet(M)$ agree if they agree on the functions and on the exact $1$-forms. On a function $f$ the right-hand side is $\iota_X df = Xf = \mathcal{L}_Xf$. On an exact form $df$ the left-hand side is $d(Xf)$ and the right-hand side is $\iota_X d(df) + d\iota_X df = 0 + d(Xf)$, which agrees. Hence the two derivations coincide, since every form is generated by the functions and the exact $1$-forms under the wedge product.

Consequences

Corollary. The following identities hold.

(a) $[\mathcal{L}_X, d] = 0$, the commutation with the exterior derivative, recovered from the formula because $d^2 = 0$.

(b) $[\mathcal{L}_X, \iota_Y] = \iota_{[X,Y]}$.

(c) $[\mathcal{L}_X, \mathcal{L}_Y] = \mathcal{L}_{[X,Y]}$.

(d) $\mathcal{L}_X(\alpha\wedge\beta) = \mathcal{L}_X\alpha\wedge\beta + \alpha\wedge\mathcal{L}_X\beta$, the Leibniz rule for the wedge product.

Proof. Part (a) is the previous proposition. For (b), both $[\mathcal{L}_X,\iota_Y]$ and $\iota_{[X,Y]}$ are graded derivations of degree $-1$ of the algebra of forms, so it suffices to compare them on the functions and on the exact $1$-forms. On a function $f$ both give $0$, since $\iota$ annihilates functions. On $df$,

$$ [\mathcal{L}_X,\iota_Y]df = \mathcal{L}_X(Yf) - \iota_Yd(Xf) = X(Yf) - Y(Xf) = [X,Y]f = \iota_{[X,Y]}df, $$

and a degree $-1$ derivation is determined by its values on the functions and the exact $1$-forms, because a $1$-form is locally a sum of terms $f\,dg$ and the derivation rule reconstructs it. Hence the two agree. For (c), $[\mathcal{L}_X,\mathcal{L}_Y] = [\mathcal{L}_X, d\iota_Y+\iota_Yd] = d[\mathcal{L}_X,\iota_Y] + [\mathcal{L}_X,\iota_Y]d = d\iota_{[X,Y]}+\iota_{[X,Y]}d = \mathcal{L}_{[X,Y]}$, using $[\mathcal{L}_X,d]=0$, part (b) and Cartan's formula. Part (d) is the derivation property.

Corollary (divergence). If $\mathrm{vol}$ is a volume form on an oriented manifold, there is a unique function $\operatorname{div}X$, the divergence, with

$$ \mathcal{L}_X\mathrm{vol} = (\operatorname{div}X)\,\mathrm{vol}, $$

and for a top form $\mathcal{L}_X\mathrm{vol} = d\iota_X\mathrm{vol}$, so $\operatorname{div}X$ is the coefficient of $d\iota_X\mathrm{vol}$ in $\mathrm{vol}$. On a Riemannian manifold with the metric volume form, $\operatorname{div}$ is the divergence of the vector calculus, and it is minus the codifferential of the $1$-form dual to $X$, as in Differential Forms and Stokes' Theorem.

Proof. The top-degree part of $\mathcal{L}_X\mathrm{vol}$ is a function multiple of $\mathrm{vol}$, which defines $\operatorname{div}X$; Cartan's formula expresses it as $d\iota_X\mathrm{vol}$, since $d\mathrm{vol} = 0$ in the top degree. The identification with the metric divergence and the codifferential follows from the definition of the latter in Differential Forms and Stokes' Theorem.

The Lie Derivative and the Covariant Derivative

The Lie derivative is defined by the bracket alone and the covariant derivative by a connection; when a connection is present their difference is tensorial and is built from the derivative of the field. The following proposition is the precise statement, and it is the one the metric case uses.

Proposition. Let $\nabla$ be a torsion-free connection on $TM$ and let $\alpha \in \Omega^1(M)$. Then

$$ (\mathcal{L}_X\alpha)(Y) = (\nabla_X\alpha)(Y) + \alpha(\nabla_YX) \qquad \text{for all } Y \in \mathrm{X}(M). $$

Equivalently, $\mathcal{L}_X = \nabla_X + \iota_{\nabla X}$ on $1$-forms, where $\nabla X$ is the $(1,1)$-tensor $\nabla X(Y,\alpha) = \alpha(\nabla_YX)$. For a general tensor field the difference $\mathcal{L}_XT - \nabla_XT$ is a tensor built from $\nabla X$ by the same Leibniz rule applied to each slot.

Proof. Since $\nabla$ is torsion-free, $[X,Y] = \nabla_XY - \nabla_YX$. Compute

$$ (\mathcal{L}_X\alpha)(Y) = X(\alpha(Y)) - \alpha([X,Y]) = (\nabla_X\alpha)(Y) + \alpha(\nabla_XY) - \alpha(\nabla_XY - \nabla_YX), $$

using $\nabla_X(\alpha(Y)) = (\nabla_X\alpha)(Y) + \alpha(\nabla_XY)$ for the metric-compatible pairing between a form and a field; the two middle terms cancel and leave $\alpha(\nabla_YX)$. The tensor statement follows by applying the same computation to each slot and using the Leibniz rule.

Remark. A vector field with $\mathcal{L}_Xg = 0$ for a Riemannian metric $g$ is a Killing field, and the identity above shows that this is the same as $\nabla X$ being skew-adjoint, equivalently $\nabla_YX$ orthogonal to $Y$ for all $Y$; the Killing fields, the isometries they generate and the geometry of the symmetry orbits are Part IV's, and the operator content is the identity displayed here. A vector field with $\mathcal{L}_X\omega = 0$ for a closed $2$-form $\omega$ is a symmetry of that form, and the whole family of such symmetry operators is the Hamiltonian machinery of Lagrangian and Hamiltonian Systems, later in this Part.

Summary

The Lie derivative along a vector field $X$ is the derivative of the pullback along the flow, $\mathcal{L}_X = \frac{d}{dt}|_{t=0}\phi_t^*$, read on functions as $Xf$, on vector fields as the bracket $[X,Y]$, and on forms by the same flow. It needs no connection and no metric; it is the derivative of the smooth structure alone. It is a degree-zero derivation of the graded algebra of forms, it commutes with the exterior derivative, and it satisfies the Leibniz rule for the wedge product.

Cartan's formula $\mathcal{L}_X = d\iota_X + \iota_Xd$ expresses it through the interior product, the degree $-1$ derivation contracting a form with the field. The formula gives the identities $[\mathcal{L}_X,d]=0$, $[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$ and $[\mathcal{L}_X,\mathcal{L}_Y]=\mathcal{L}_{[X,Y]}$; on a volume form it produces the divergence, $\mathcal{L}_X\mathrm{vol} = (\operatorname{div}X)\mathrm{vol}$. Where a torsion-free connection is present, $\mathcal{L}_X = \nabla_X + \iota_{\nabla X}$ on the forms, so the Lie derivative differs from the covariant derivative by the tensor $\nabla X$, and the vanishing of $\mathcal{L}_Xg$ is the infinitesimal isometry condition of the metric geometry of Part IV.

Summary of Notation

Symbol	Meaning
$X, Y, Z$	Vector fields in $\mathrm{X}(M)$
$\phi_t$, $\phi : (-\epsilon,\epsilon)\times U \to M$	Local flow of $X$; $\phi_s\circ\phi_t = \phi_{s+t}$
$\mathcal{L}_X$	Lie derivative along $X$
$\mathcal{L}_Xf = Xf = df(X)$	Lie derivative of a function
$\mathcal{L}_XY = [X,Y]$	Lie derivative of a vector field; the Lie bracket
$\mathcal{L}_X\omega = \frac{d}{dt}\big\|_0\phi_t^*\omega$	Lie derivative of a form by the flow
$\mathcal{L}_X(\alpha\wedge\beta) = \mathcal{L}_X\alpha\wedge\beta + \alpha\wedge\mathcal{L}_X\beta$	Leibniz rule; $\mathcal{L}_X$ is a degree-zero derivation
$[\mathcal{L}_X,d]=0$	Commutation with the exterior derivative
$\iota_X$	Interior product, a degree $-1$ derivation; $\iota_X^2=0$
$\mathcal{L}_X = d\iota_X + \iota_X d$	Cartan's formula
$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$	Commutation of the Lie derivative and the contraction
$[\mathcal{L}_X,\mathcal{L}_Y]=\mathcal{L}_{[X,Y]}$	The Lie derivative is a representation of the Lie algebra of fields
$\operatorname{div}X$, $\mathcal{L}_X\mathrm{vol} = (\operatorname{div}X)\mathrm{vol}$	Divergence from the volume form
$(\mathcal{L}_X\alpha)(Y) = (\nabla_X\alpha)(Y)+\alpha(\nabla_YX)$	Relation to a torsion-free covariant derivative
$\nabla X$, $\iota_{\nabla X}$	The $(1,1)$-tensor $\nabla X(Y,\alpha)=\alpha(\nabla_YX)$; its contraction
Killing field	$\mathcal{L}_Xg=0$; named here, geometry of Part IV