The Lie Derivative of a Tensor Field
The
Lie derivative
measures the rate of change of a vector field along the flow of another, by comparing the field at
nearby times after transporting it back to a common point. The same comparison applies to a covariant
tensor field, with one simplification: a covariant tensor is transported by pulling back along the
flow, an operation that exists for every smooth map and needs no inverse. This makes the Lie
derivative of a covariant tensor field both natural and computable.
Let \(M\) be a smooth manifold and \(V\) a smooth vector field on \(M\), with
flow
\(\theta\). The discussion is carried out for a manifold without boundary; it carries over unchanged
to a manifold with boundary provided \(V\) is tangent to the boundary, so that its flow exists. For
any point \(p\) and any time \(t\) small enough, \(\theta_t\) is a diffeomorphism from a neighborhood
of \(p\) onto a neighborhood of \(\theta_t(p)\), so its differential pulls back tensors at
\(\theta_t(p)\) to tensors at \(p\). Applying this to the value of a tensor field \(A\) at
\(\theta_t(p)\) gives a curve of tensors at the single point \(p\), and differentiating that curve
isolates the infinitesimal change of \(A\) along the flow.
Definition: Lie Derivative of a Covariant Tensor Field
Let \(V\) be a smooth vector field on a smooth manifold \(M\), with flow \(\theta\), and let
\(A\) be a smooth
covariant tensor field
on \(M\). The Lie derivative of \(A\) with respect to \(V\), denoted
\(\mathcal{L}_V A\), is the covariant tensor field whose value at each point \(p\) is
\[
(\mathcal{L}_V A)_p
= \left. \frac{d}{dt} \right|_{t=0} \big(\theta_t^* A\big)_p
= \lim_{t \to 0}
\frac{\big(d(\theta_t)_p^*\, A_{\theta_t(p)}\big) - A_p}{t} ,
\]
provided the derivative exists. Here \(\theta_t^* A\) is the
pullback
of \(A\) by the diffeomorphism \(\theta_t\), defined wherever \(\theta_t\) is, and
\(d(\theta_t)_p^*\, A_{\theta_t(p)}\) is its value at \(p\).
The difference quotient is well posed because both terms are covariant \(k\)-tensors at the same
point \(p\), elements of a fixed vector space, so their difference and the limit make sense. This is
the feature that makes the comparison along a flow possible at all: without transporting the value at
\(\theta_t(p)\) back to \(p\), the two tensors would live in different spaces. That the limit exists
and varies smoothly is the content of the next result.
Theorem: Existence and Smoothness of the Lie Derivative
With \(M\), \(V\), and \(A\) as in the definition, the derivative defining
\((\mathcal{L}_V A)_p\) exists for every \(p \in M\), and \(\mathcal{L}_V A\) is a smooth
covariant tensor field on \(M\).
Proof:
The expression being differentiated lies in the fixed vector space of covariant \(k\)-tensors at
\(p\) for every \(t\) in an interval around \(0\), so the derivative, if it exists, is again such
a tensor. To establish existence and smoothness simultaneously, work in smooth local
coordinates. The pullback \(\theta_t^* A\) has component functions obtained by composing the
components of \(A\) with \(\theta_t\) and with the partial derivatives of the flow in the
coordinate directions; both depend smoothly on \((t, p)\) jointly, because the flow is a smooth
function of time and position by the
fundamental theorem on flows.
A function smooth in \((t, p)\) has a partial derivative in \(t\) that exists and is smooth in
\(p\); applying this to each component function shows that the limit defining
\((\mathcal{L}_V A)_p\) exists and that the resulting components are smooth. By the smoothness
criteria for tensor fields, \(\mathcal{L}_V A\) is a smooth covariant tensor field.
The construction reduces to the familiar one in low rank. For a \(0\)-tensor field, a smooth
function, the pullback is composition with the flow and the Lie derivative is the ordinary derivative
of the function along the flow, recovering the directional derivative. For higher rank the same
definition organizes the change of a field of multilinear objects, and the rules for computing it,
developed next, mirror the product rule of calculus.
The Product Rule
Differentiation along a flow obeys a Leibniz rule with respect to every product that combines tensor
fields. Because the Lie derivative is a limit of difference quotients of pullbacks, and the pullback
distributes over functions, tensor products, and evaluation on vector fields, each such product
inherits a product rule. These rules reduce every Lie-derivative computation to the case of functions
and coordinate differentials.
Theorem: Product Rules for the Lie Derivative
Let \(M\) be a smooth manifold and \(V\) a smooth vector field on \(M\). Let \(f\) be a smooth
function, regarded as a \(0\)-tensor field, and let \(A, B\) be smooth covariant tensor fields.
(a) The Lie derivative of a function is its directional derivative along \(V\):
\[
\mathcal{L}_V f = V f .
\]
(b) For a function times a tensor field,
\[
\mathcal{L}_V (f A) = (\mathcal{L}_V f)\, A + f\, \mathcal{L}_V A .
\]
(c) For a tensor product,
\[
\mathcal{L}_V (A \otimes B) = (\mathcal{L}_V A) \otimes B + A \otimes \mathcal{L}_V B .
\]
(d) If \(A\) is a smooth covariant \(k\)-tensor field and \(X_1, \ldots, X_k\) are smooth vector
fields, then the smooth function \(A(X_1, \ldots, X_k)\) satisfies
\[
\mathcal{L}_V\big(A(X_1, \ldots, X_k)\big)
= (\mathcal{L}_V A)(X_1, \ldots, X_k)
+ \sum_{i=1}^{k} A\big(X_1, \ldots, \mathcal{L}_V X_i, \ldots, X_k\big) ,
\]
where \(\mathcal{L}_V X_i\) is the
Lie derivative
of the vector field \(X_i\).
Proof:
Let \(\theta\) be the flow of \(V\). For part (a), the pullback of a function by \(\theta_t\) is
the composition \(\theta_t^* f = f \circ \theta_t\), so at a point \(p\),
\[
(\mathcal{L}_V f)(p)
= \left. \frac{d}{dt} \right|_{t=0} f\big(\theta_t(p)\big) .
\]
The curve \(t \mapsto \theta_t(p)\) is the
integral curve
of \(V\) starting at \(p\), and its velocity at \(t = 0\) is \(V_p\). Differentiating the
composite function therefore gives \(df_p(V_p) = V_p f\), which is \((Vf)(p)\). Since \(p\) was
arbitrary, \(\mathcal{L}_V f = V f\).
Parts (b) and (c) follow from the same pattern used for the Lie derivative of vector fields. The
pullback is multiplicative: \(\theta_t^*(fA) = (\theta_t^* f)(\theta_t^* A)\) and
\(\theta_t^*(A \otimes B) = (\theta_t^* A) \otimes (\theta_t^* B)\), by the pullback properties of
tensor fields. Each identity expresses a \(t\)-dependent product of two factors that both equal
their undifferentiated values at \(t = 0\). Differentiating a product of two such smooth curves
of tensors at \(t = 0\) gives the ordinary product rule, the derivative of the first factor times
the second at \(t = 0\) plus the first at \(t = 0\) times the derivative of the second. For (b)
this reads \((\mathcal{L}_V f) A + f\, \mathcal{L}_V A\), and for (c) it reads
\((\mathcal{L}_V A) \otimes B + A \otimes \mathcal{L}_V B\).
For part (d), the function \(A(X_1, \ldots, X_k)\) is obtained from \(A\) and the vector fields
by a contraction that, under pullback by the diffeomorphism \(\theta_t\), satisfies
\[
\theta_t^*\big(A(X_1, \ldots, X_k)\big)
= (\theta_t^* A)\big(\theta_t^* X_1, \ldots, \theta_t^* X_k\big) ,
\]
where \(\theta_t^* X_i\) is the pullback of the vector field \(X_i\) by \(\theta_t\), with value
\(d(\theta_{-t})_{\theta_t(p)}\big( (X_i)_{\theta_t(p)} \big)\) at \(p\), exactly the transport
used in the Lie derivative of a vector field. The left side is a function whose ordinary
\(t\)-derivative at \(0\) is the left side of
the claimed identity, by part (a). The right side is a value of a multilinear expression in
\(k + 1\) smooth curves of objects, namely \(\theta_t^* A\) and the \(\theta_t^* X_i\), each
equal to its undifferentiated value at \(t = 0\). Differentiating this multilinear expression at
\(t = 0\) produces one term for each factor, the factor's derivative with the others held at
their \(t = 0\) values: the \(A\)-term gives \((\mathcal{L}_V A)(X_1, \ldots, X_k)\), and the
\(i\)-th vector-field term gives \(A(X_1, \ldots, \mathcal{L}_V X_i, \ldots, X_k)\), since
\(\frac{d}{dt}\big|_{0} \theta_t^* X_i = \mathcal{L}_V X_i\). Summing yields the stated formula.
Part (d) is the form most often used in practice, as it relates the Lie derivative of a tensor field
to those of the vector fields it is evaluated on, quantities that can themselves be computed as
brackets. Solving the identity for \((\mathcal{L}_V A)(X_1, \ldots, X_k)\) expresses the Lie
derivative of \(A\) entirely through directional derivatives of functions and Lie brackets of vector
fields, a coordinate-free computation carried out in the next section.
Computation in Coordinates
The product rule of the previous section has two immediate consequences that together make the Lie
derivative computable without ever constructing the flow. The first expresses the Lie derivative of
a tensor field acting on vector fields through Lie brackets; the second shows that the Lie derivative
commutes with the differential of a function. Combined, they reduce the coordinate computation to
differentiating component functions and the components of the vector field.
Proof:
Solve the product rule for the Lie derivative acting on vector fields for the term
\((\mathcal{L}_V A)(X_1, \ldots, X_k)\), obtaining
\[
(\mathcal{L}_V A)(X_1, \ldots, X_k)
= \mathcal{L}_V\big(A(X_1, \ldots, X_k)\big)
- \sum_{i=1}^{k} A\big(X_1, \ldots, \mathcal{L}_V X_i, \ldots, X_k\big) .
\]
The function \(A(X_1, \ldots, X_k)\) has Lie derivative equal to its directional derivative
\(V(A(X_1, \ldots, X_k))\), and the Lie derivative of a vector field is the bracket,
\(\mathcal{L}_V X_i = [V, X_i]\). Substituting both gives the stated formula.
Theorem: The Lie Derivative Commutes with the Differential
For any smooth function \(f\) on \(M\) and any smooth vector field \(V\),
\[
\mathcal{L}_V (df) = d(\mathcal{L}_V f) = d(V f) .
\]
Proof:
Both sides are covector fields, so it suffices to show they agree on an arbitrary smooth vector
field \(X\). Apply the bracket formula above to the covariant \(1\)-tensor field
\(df\),
using \(df(X) = Xf\):
\[
(\mathcal{L}_V\, df)(X)
= V\big(df(X)\big) - df\big([V, X]\big)
= V(Xf) - [V, X]f .
\]
The bracket of vector fields acts on functions by \([V, X]f = V(Xf) - X(Vf)\), so the right side
equals \(V(Xf) - (V(Xf) - X(Vf)) = X(Vf)\). On the other hand
\(d(Vf)(X) = X(Vf)\). The two sides agree on every \(X\), hence
\(\mathcal{L}_V(df) = d(Vf)\).
A coordinate computation
These identities yield a direct method in coordinates. Any tensor field is a sum of component
functions times tensor products of coordinate differentials, and each coordinate differential is the
differential of a coordinate function, so the Lie derivative of \(dx^i\) is
\(\mathcal{L}_V(dx^i) = d(\mathcal{L}_V x^i) = d(V x^i) = dV^i\), where \(V^i = V x^i\) is the
\(i\)-th component of \(V\). The product rule then expands the Lie derivative term by term.
Take a smooth covariant \(2\)-tensor field written in
coordinates
as \(A = A_{ij}\, dx^i \otimes dx^j\). Applying the product rule for a function times a tensor and for
a tensor product,
\[
\begin{align*}
\mathcal{L}_V A
&= \mathcal{L}_V\big(A_{ij}\, dx^i \otimes dx^j\big) \\\\
&= (\mathcal{L}_V A_{ij})\, dx^i \otimes dx^j
+ A_{ij}\,(\mathcal{L}_V dx^i) \otimes dx^j
+ A_{ij}\, dx^i \otimes (\mathcal{L}_V dx^j) \\\\
&= (V A_{ij})\, dx^i \otimes dx^j
+ A_{ij}\, dV^i \otimes dx^j
+ A_{ij}\, dx^i \otimes dV^j .
\end{align*}
\]
Expanding \(dV^i = (\partial V^i / \partial x^k)\, dx^k\) and relabeling summation indices collects
everything onto the basis \(dx^i \otimes dx^j\), giving the component formula
\[
\mathcal{L}_V A
= \left(
V A_{ij}
+ A_{kj}\, \frac{\partial V^k}{\partial x^i}
+ A_{ik}\, \frac{\partial V^k}{\partial x^j}
\right) dx^i \otimes dx^j .
\]
The first term differentiates the components along \(V\); the remaining two account for the change of
the coordinate differentials under the flow, one for each tensor slot. The pattern generalizes
directly to higher rank: the Lie derivative of a covariant \(k\)-tensor field has a leading term
\(V A_{i_1 \cdots i_k}\) and one correction term \(A_{\cdots k \cdots}\,\partial V^k / \partial
x^{i_j}\) for each of the \(k\) slots.
Invariance and Killing Fields
A vector field has vanishing Lie derivative along \(V\) exactly when it is
invariant
under the flow of \(V\). The Lie derivative of a covariant tensor field has the same interpretation:
it vanishes precisely when the tensor field is carried to itself by the flow. This turns a
differential condition into a geometric symmetry, and it is the condition that singles out the
symmetries of a geometric structure.
Definition: Flow-Invariant Tensor Field
Let \(A\) be a smooth covariant tensor field on \(M\) and let \(\theta\) be a flow on \(M\). The
tensor field \(A\) is invariant under \(\theta\) if it is unchanged by pulling
back along the flow wherever the flow is defined:
\[
\big(\theta_t\big)^* A = A
\qquad \text{wherever } \theta_t \text{ is defined.}
\]
Spelled out pointwise, this is the requirement that
\(d(\theta_t)_p^*\big(A_{\theta_t(p)}\big) = A_p\) for every \((t, p)\) in the domain of
\(\theta\). When \(\theta\) is the flow of a vector field \(V\), invariance under \(\theta\) is
also expressed by saying \(A\) is invariant under \(V\).
Vanishing of the Lie derivative is the infinitesimal form of this invariance. Establishing the
equivalence requires differentiating the pullback at times other than zero, which the following
extends from the defining derivative at \(t = 0\).
Theorem: Vanishing Lie Derivative and Invariance
Let \(M\) be a smooth manifold and \(V \in \mathfrak{X}(M)\). A smooth covariant tensor field
\(A\) is invariant under the flow of \(V\) if and only if \(\mathcal{L}_V A = 0\).
Proof:
Let \(\theta\) be the flow of \(V\). The essential computation is the value of the \(t\)-derivative
of the pullback at an arbitrary time \(t_0\) in the domain, not only at \(0\). Using the group law
\(\theta_{t} = \theta_{(t - t_0)} \circ \theta_{t_0}\) and the composition rule for pullbacks, a
change of variable \(s = t - t_0\) gives
\[
\left. \frac{d}{dt} \right|_{t = t_0} \big(\theta_t^* A\big)_p
= \left. \frac{d}{ds} \right|_{s = 0}
\big(\theta_{t_0}^*\, \theta_s^* A\big)_p
= \big(\theta_{t_0}^*\, \mathcal{L}_V A\big)_p ,
\]
where the pullback by the fixed diffeomorphism \(\theta_{t_0}\) passes outside the
\(s\)-derivative because it is linear and \(t_0\)-constant, and the inner derivative at
\(s = 0\) is the Lie derivative by definition.
Suppose first that \(A\) is invariant, so \(\theta_t^* A = A\) for all \(t\) in the domain. Then
the curve \(t \mapsto (\theta_t^* A)_p\) is constant, its derivative at \(t = 0\) is zero, and
that derivative is \((\mathcal{L}_V A)_p\); since \(p\) is arbitrary, \(\mathcal{L}_V A = 0\).
Conversely, suppose \(\mathcal{L}_V A = 0\). By the displayed identity, the derivative of
\(t \mapsto (\theta_t^* A)_p\) vanishes at every \(t_0\) in the domain, because it equals
\((\theta_{t_0}^*\, \mathcal{L}_V A)_p = 0\). A curve in a fixed vector space with identically
zero derivative on an interval is constant, so \((\theta_t^* A)_p = (\theta_0^* A)_p = A_p\) for
all \(t\). As \(p\) ranges over \(M\), this is the invariance \(\theta_t^* A = A\).
The result completes the parallel with vector fields: a vector field is invariant under the flow of
\(V\) if and only if its Lie derivative along \(V\) vanishes, and a covariant tensor field obeys the
identical criterion. The condition \(\mathcal{L}_V A = 0\) thus identifies the infinitesimal
symmetries of \(A\), the directions along which the tensor field does not change.
The symmetries of a metric
The case that drives much of geometry is a
symmetric
covariant \(2\)-tensor field that is positive definite at each point, a field of inner products
assigning a length to every tangent vector. A vector field \(V\) along which such a field has
vanishing Lie derivative generates a flow that preserves lengths and angles: an infinitesimal
rigid motion of the geometry. These distinguished vector fields are the infinitesimal isometries,
and the condition \(\mathcal{L}_V g = 0\) on the field of inner products \(g\) is the equation
that defines them. Written in coordinates by the component formula of the previous section, it
becomes a first-order linear system on the components of \(V\), and its solutions encode the
continuous symmetries of the space. A vector field satisfying \(\mathcal{L}_V g = 0\) for such a
field of inner products is called a Killing field; its systematic study, and the
precise notion of the metric \(g\) it presupposes, belong to the development of Riemannian
geometry that this algebraic machinery has been built to support. The study of when such
symmetries exist, how many there can be, and what structure they form is one of the destinations
toward which the machinery of tensor fields has been built.