The Lie Derivative of a Vector Field
Differentiating a smooth real-valued function on a manifold along a tangent direction
is already settled. A
vector field
\(V\) acts on \(f \in C^\infty(M)\) to produce another smooth function \(Vf\), and at
each point \(p\) the number \((Vf)(p)\) is the directional derivative of \(f\) in the
direction prescribed by \(V_p\). Differentiating a vector field along a tangent
direction is a strictly harder problem. The difficulty has nothing to do with
smoothness, and is instead a structural obstruction concerning where the tangent
vectors live.
The Difficulty: Tangent Vectors at Different Points
In Euclidean space the directional derivative of a smooth vector field
\(W\) on an open subset of \(\mathbb{R}^n\) in the direction of a vector
\(v \in T_p \mathbb{R}^n\) is the limit
\[
\begin{align*}
D_v W(p)
&= \frac{d}{dt}\bigg|_{t = 0} W_{p + tv} \\\\
&= \lim_{t \to 0} \frac{W_{p + tv} - W_p}{t}.
\end{align*}
\]
The expression makes sense as written because tangent spaces at different points of
\(\mathbb{R}^n\) are canonically identified with \(\mathbb{R}^n\) itself, so
\(W_{p + tv}\) and \(W_p\) are both elements of the same vector space and their
difference can be formed.
On a general smooth manifold this identification is no longer available. Even after
replacing the straight line \(p + tv\) by a smooth curve \(\gamma\) with
\(\gamma(0) = p\) and
\(\gamma'(0) = v\), the vectors \(W_{\gamma(t)} \in T_{\gamma(t)} M\) and
\(W_{\gamma(0)} \in T_{\gamma(0)} M\) live in genuinely different vector spaces, and
no canonical isomorphism between them is available. The naive difference quotient
\(\bigl( W_{\gamma(t)} - W_{\gamma(0)} \bigr) / t\) is therefore not even
a well-defined element of any single vector space. The construction underlying the
Euclidean directional derivative breaks down at the most basic level.
Replacing the bare tangent vector \(v\) with a vector field \(V\) restores enough
structure to repair the construction. A vector field carries with it its
flow
\(\theta\), and the flow supplies a family of diffeomorphisms \(\theta_t\) defined on
open subsets of \(M\), each of which carries tangent spaces at one point onto tangent
spaces at another by its differential. Using the inverse \(\theta_{-t}\) of
\(\theta_t\), the vector \(W_{\theta_t(p)} \in T_{\theta_t(p)} M\) can be transported
back to \(T_p M\) as \(d(\theta_{-t})_{\theta_t(p)}\bigl(W_{\theta_t(p)}\bigr)\), and
this transported vector and the original \(W_p\) are now both elements of \(T_p M\).
Their difference makes sense, and the limit of the difference quotient, provided it
exists, is the construction we have been seeking.
The Definition
Definition: Lie Derivative of a Vector Field
Let \(M\) be a smooth manifold, let \(V \in \mathfrak{X}(M)\) be a smooth vector
field with flow \(\theta\), and let \(W \in \mathfrak{X}(M)\) be a smooth vector
field. The Lie derivative of \(W\) with respect to \(V\) at a
point \(p \in M\) is the element of \(T_p M\) defined by
\[
\begin{align*}
(\mathcal{L}_V W)_p
&= \frac{d}{dt}\bigg|_{t = 0}
d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr) \\\\
&= \lim_{t \to 0}
\frac{d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr) - W_p}{t},
\end{align*}
\]
provided the derivative on the right exists.
The assignment \(p \mapsto (\mathcal{L}_V W)_p\) defines a
rough vector field
on \(M\), denoted \(\mathcal{L}_V W\).
The difference quotient inside the limit is a genuine element of \(T_p M\) for every
sufficiently small \(t \neq 0\). The diffeomorphism \(\theta_t\) is defined on a
neighborhood of \(p\) for small \(t\), and its inverse \(\theta_{-t}\) carries
\(\theta_t(p)\) back to \(p\). The differential
\(d(\theta_{-t})_{\theta_t(p)} : T_{\theta_t(p)} M \to T_p M\) is a linear isomorphism,
and its image
\(d(\theta_{-t})_{\theta_t(p)}\bigl(W_{\theta_t(p)}\bigr)\) lies in \(T_p M\), the
same vector space as \(W_p\). What the definition asserts to exist is the
\(t\)-derivative of this transported family at \(t = 0\). Existence and smoothness of
this derivative are exactly the content of the lemma below.
Existence and Smoothness
Although the rough vector field \(\mathcal{L}_V W\) is defined point by point as a
derivative that might in principle fail to exist, in fact the derivative exists at
every point and the assignment is smooth. The proof works in coordinates. The
differential \(d(\theta_{-t})_{\theta_t(p)}\) has an explicit matrix in any chart, and
its entries are smooth functions of \((t, p)\). The \(t\)-derivative at \(t = 0\) of a
smooth function of \((t, p)\) is itself a smooth function of \(p\), so the Lie
derivative exists pointwise and varies smoothly.
Lemma (Existence and Smoothness of the Lie Derivative)
Let \(M\) be a smooth manifold and \(V, W \in \mathfrak{X}(M)\). The Lie
derivative \((\mathcal{L}_V W)_p\) exists at every \(p \in M\), and the resulting
assignment \(\mathcal{L}_V W\) is a smooth vector field on \(M\).
Proof (outline):
Fix \(p \in M\) and a smooth chart \((U, (x^i))\) containing \(p\). Openness of
the
flow domain
gives an open interval \(J_0\) containing \(0\) and an open neighborhood
\(U_0 \subseteq U\) of \(p\) such that the restricted flow
\(\theta : J_0 \times U_0 \to U\) is smooth and takes values in the chart domain.
Write the component functions of \(\theta\) in the chart as
\(\theta(t, x) = \bigl(\theta^1(t, x), \ldots, \theta^n(t, x)\bigr)\).
The differential
\(d(\theta_{-t})_{\theta_t(x)} : T_{\theta_t(x)} M \to T_x M\) is the differential
of the map \(\theta_{-t}\) at the point \(\theta_t(x)\). In the coordinate basis,
its matrix is the Jacobian of \(\theta_{-t}\) evaluated there, namely
\[
\left( \frac{\partial \theta^i}{\partial x^j}\bigl(-t, \, \theta(t, x)\bigr) \right) .
\]
Applying this differential to the vector
\(W_{\theta_t(x)} = W^j(\theta(t, x)) \, \partial/\partial x^j |_{\theta_t(x)}\) gives
\[
d(\theta_{-t})_{\theta_t(x)}\bigl( W_{\theta_t(x)} \bigr)
= \frac{\partial \theta^i}{\partial x^j}\bigl(-t, \, \theta(t, x)\bigr)
\, W^j(\theta(t, x))
\, \frac{\partial}{\partial x^i}\bigg|_x .
\]
The coefficient of each basis vector \(\partial/\partial x^i |_x\) is a smooth
function of \((t, x) \in J_0 \times U_0\), being built by composition,
differentiation, and multiplication of the smooth maps \(\theta\) and \(W^j\).
For a smooth function of \((t, x)\), the \(t\)-derivative at \(t = 0\) exists and
is itself smooth in \(x\). Each coefficient of the basis decomposition therefore
has a well-defined \(t\)-derivative at \(t = 0\), and the resulting vector
\[
(\mathcal{L}_V W)_x
= \frac{d}{dt}\bigg|_{t = 0}
d(\theta_{-t})_{\theta_t(x)}\bigl( W_{\theta_t(x)} \bigr)
\in T_x M
\]
depends smoothly on \(x \in U_0\). The argument is independent of \(p\) and of
the chart, so the Lie derivative exists at every point of \(M\) and is smooth in
a neighborhood of each point.
The definition gives a way to differentiate one vector field along the flow of
another, but it is computationally awkward. Evaluating the Lie derivative through the
formula above requires knowing the flow \(\theta\) explicitly, which is rarely
possible. The central theorem of the next section identifies \(\mathcal{L}_V W\) with
an object already in our hands. That object is the
Lie bracket
\([V, W]\) of the two vector fields, and the identification makes the Lie derivative
computable without ever writing down a flow. The identity \(\mathcal{L}_V W = [V, W]\) is the
bridge between the geometric content of the bracket (rate of change along a flow) and
the algebraic content already established for it on the page treating vector fields
under smooth maps.
Lie Derivative Equals Lie Bracket
The
Lie bracket
\([V, W]\) of two smooth vector fields was constructed earlier through their action
on smooth functions, as the vector field whose action satisfies
\([V, W] f = V(W f) - W(V f)\). That construction was algebraic, since it referred only
to the derivations attached to \(V\) and \(W\). The geometric meaning of the resulting
field was deferred to a later stage. The theorem below settles the question in full.
The bracket of \(V\) and \(W\) is exactly the Lie derivative of \(W\) along the flow
of \(V\). Two superficially different objects turn out to coincide as vector fields on
the manifold. One is defined by an algebraic commutator, and the other by a limit
involving the flow.
Theorem (Lie Derivative Equals Lie Bracket)
Let \(M\) be a smooth manifold and let \(V, W \in \mathfrak{X}(M)\). Then
\[
\mathcal{L}_V W = [V, W] .
\]
The proof distinguishes three cases according to the position of the point \(p\)
relative to the
support
of \(V\). The set of
regular points
\(\mathcal{R}(V) = \{p \in M : V_p \neq 0\}\) is open in \(M\) by continuity, and its
closure is exactly \(\mathrm{supp}\,V\). The complement of the support is the set on
which \(V\) vanishes identically in a neighborhood. The boundary between these two
open sets is the regular-singular boundary, where both quantities will be controlled
by continuity.
Proof:
Fix \(p \in M\). The three cases are: \(p \in \mathcal{R}(V)\), \(p \in
\mathrm{supp}\,V \setminus \mathcal{R}(V)\), and \(p \in M \setminus
\mathrm{supp}\,V\). Together they exhaust \(M\), since
\(\mathrm{supp}\,V = \overline{\mathcal{R}(V)}\).
Case 1: \(p \in \mathcal{R}(V)\).
The
canonical form theorem
applied at \(p\) provides smooth coordinates \((u^1, \ldots, u^n)\) on some
neighborhood of \(p\) in which \(V\) has the coordinate representation
\(V = \partial / \partial u^1\). In these coordinates the flow of \(V\) is simply
translation in the first coordinate,
\[
\theta_t(u^1, u^2, \ldots, u^n) = (u^1 + t, \, u^2, \ldots, u^n) ,
\]
for every \((t, u)\) in some neighborhood of \((0, p)\). The differential
\(d(\theta_{-t})_{\theta_t(u)}\) of a translation by a constant is the identity
on tangent spaces. Every coordinate basis vector is sent to the corresponding
coordinate basis vector at the image point. Writing
\(W = W^j(u) \, \partial / \partial u^j\) in these coordinates, the transported
vector at the point \(u\) is
\[
d(\theta_{-t})_{\theta_t(u)}\bigl( W_{\theta_t(u)} \bigr)
= W^j(u^1 + t, u^2, \ldots, u^n) \, \frac{\partial}{\partial u^j}\bigg|_u .
\]
Differentiating in \(t\) at \(t = 0\) and using the chain rule gives the Lie
derivative in coordinates:
\[
(\mathcal{L}_V W)_u
= \frac{\partial W^j}{\partial u^1}(u)
\, \frac{\partial}{\partial u^j}\bigg|_u .
\]
On the other hand, the
coordinate formula for the Lie bracket
applied to
\(V = \partial / \partial u^1\) and \(W = W^j \partial / \partial u^j\) yields
\[
[V, W]_u^j
= V^i \, \frac{\partial W^j}{\partial u^i}(u)
- W^i \, \frac{\partial V^j}{\partial u^i}(u) ,
\]
and the components of \(V\) are \(V^i = \delta^i_1\), the Kronecker delta. The
first term reduces to \(V^i \partial W^j / \partial u^i = \partial W^j /
\partial u^1\), since the sum over \(i\) is killed by the delta. The second
term involves \(\partial V^j / \partial u^i = \partial (\delta^j_1) /
\partial u^i\), which vanishes identically because each component of \(V\) is
constant. The bracket therefore reduces to
\[
[V, W]_u
= \frac{\partial W^j}{\partial u^1}(u) \, \frac{\partial}{\partial u^j}\bigg|_u ,
\]
which is exactly the expression obtained for \((\mathcal{L}_V W)_u\). The two
agree at every \(p \in \mathcal{R}(V)\).
Case 2: \(p \in \mathrm{supp}\,V \setminus \mathcal{R}(V)\).
By the definition of support and the openness of \(\mathcal{R}(V)\), the set
\(\mathcal{R}(V)\) is dense in its closure \(\mathrm{supp}\,V\). In particular,
every neighborhood of \(p\) meets \(\mathcal{R}(V)\). Both \(\mathcal{L}_V W\)
and \([V, W]\) are smooth vector fields on \(M\), so each is a continuous map from
\(M\) into \(TM\). For \(\mathcal{L}_V W\) this is the existence-and-smoothness lemma
of the previous section, and for \([V, W]\) it is the smoothness of the Lie bracket of
two smooth vector fields. Two continuous maps into a Hausdorff space agree on
the closure of any set on which they agree, and the tangent bundle \(TM\)
is Hausdorff. Case 1
established \(\mathcal{L}_V W = [V, W]\) on the open dense set
\(\mathcal{R}(V) \subseteq \mathrm{supp}\,V\), so the equality extends to all
of \(\mathrm{supp}\,V\), and in particular holds at the point \(p\).
Case 3: \(p \in M \setminus \mathrm{supp}\,V\).
By definition of
the support, \(V\) vanishes identically on some open neighborhood \(U\) of \(p\).
Every point of \(U\) is then a singular point of \(V\), so the
dichotomy for integral curves at regular and singular points
makes the trajectory through each such point constant. Hence
\(\theta_t(q) = q\) for every \(q \in U\) and every \(t\), and
\(d(\theta_{-t})_q\) is the identity on \(T_q M\). The Lie-derivative
difference quotient at \(p\) is therefore
\[
\frac{d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr) - W_p}{t}
= \frac{W_p - W_p}{t} = 0 ,
\]
so \((\mathcal{L}_V W)_p = 0\). The bracket also vanishes at \(p\). In any chart
around \(p\) contained in \(U\), the coordinate formula
\([V, W]^j = V^i \partial W^j / \partial x^i - W^i \partial V^j / \partial x^i\)
has its first term killed by \(V^i \equiv 0\) on \(U\). Its second term is killed by
\(\partial V^j / \partial x^i \equiv 0\) on \(U\), since the partial derivative of an
identically zero function vanishes. Hence
\([V, W]_p = 0\), and both sides agree at \(p\).
Each case covers part of \(M\) and the three together cover all of \(M\), so
\(\mathcal{L}_V W = [V, W]\) everywhere on \(M\).
The identification of the Lie derivative with the Lie bracket has consequences in
both directions. From left to right, the bracket, defined algebraically, now
acquires the geometric reading announced in the title of this section, namely that
\([V, W]\) is the infinitesimal rate at which \(W\) is dragged along the flow of
\(V\). From right to left, the bracket is the computational object. The
coordinate-formula bracket can be evaluated in any chart without reference to a flow,
and the identity delivers \(\mathcal{L}_V W\) at the same cost. The geometric interpretation is the
starting point for the structural results of the rest of this development. The
algebraic consequences are collected in the corollary below.
Corollary (Algebraic Properties of the Lie Derivative)
Let \(V, W, X \in \mathfrak{X}(M)\), \(g \in C^\infty(M)\), and let
\(F : M \to N\) be a diffeomorphism. The Lie derivative satisfies the following
identities.
(a) Antisymmetry:
\[
\mathcal{L}_V W = -\mathcal{L}_W V.
\]
(b) Compatibility with the bracket:
\[
\mathcal{L}_V [W, X] = [\mathcal{L}_V W, X] + [W, \mathcal{L}_V X].
\]
(c) Iterated Lie derivatives:
\[
\mathcal{L}_{[V, W]} X
= \mathcal{L}_V \mathcal{L}_W X - \mathcal{L}_W \mathcal{L}_V X.
\]
(d) Leibniz rule in the second slot:
\[
\mathcal{L}_V (g W) = (V g) \, W + g \, \mathcal{L}_V W.
\]
(e) Naturality under diffeomorphisms:
\[
F_* (\mathcal{L}_V X) = \mathcal{L}_{F_* V} F_* X.
\]
Proof:
Each identity follows from the theorem just proved and the corresponding
algebraic property of the Lie bracket. Statement (a) is the
antisymmetry of the bracket
\([V, W] = -[W, V]\) rewritten through the identity
\(\mathcal{L}_V W = [V, W]\). For statements (b) and (c), apply the Jacobi
identity \([V, [W, X]] + [W, [X, V]] + [X, [V, W]] = 0\) and use antisymmetry
to bring it into the form
\[
[V, [W, X]] = [[V, W], X] + [W, [V, X]] .
\]
Reading the left-hand side as \(\mathcal{L}_V[W, X]\) and reading the right-hand
side either as \([\mathcal{L}_V W, X] + [W, \mathcal{L}_V X]\) or as
\(\mathcal{L}_{[V, W]} X + \mathcal{L}_W \mathcal{L}_V X\) gives (b) and (c)
respectively. The second reading uses
\([W, [V, X]] = \mathcal{L}_W \mathcal{L}_V X\) and rearranges to put
\(\mathcal{L}_{[V, W]} X\) on the left. Statement (d) is the
function-linearity rule for the Lie bracket in the second argument,
\([V, gW] = (Vg) W + g [V, W]\), read through the identity. Statement (e)
is the
naturality of the Lie bracket under diffeomorphisms,
\(F_* [V, X] = [F_* V, F_* X]\).
The Adjoint Representations Reconstructed
In the matrix-Lie-group setting, the
adjoint representation
was introduced through the explicit formula \(\mathrm{Ad}_g(X) = g X g^{-1}\),
and its infinitesimal counterpart was the
matrix ad map
\(\mathrm{ad}_X(Y) = X Y - Y X\). For a general Lie group \(G\) the same two
maps exist, but their construction is no longer a matrix computation. It relies on
smooth-manifold infrastructure that is now in place. For
each \(g \in G\), the
conjugation
map \(\mathrm{conj}_g : G \to G\), \(h \mapsto g h g^{-1}\), is a
Lie group homomorphism
fixing the identity. Its differential at the identity is a linear automorphism
\(d(\mathrm{conj}_g)_e : T_e G \to T_e G\), and the
evaluation isomorphism
identifies \(T_e G\) with the
Lie algebra of \(G\).
Setting \(\mathrm{Ad}_g = d(\mathrm{conj}_g)_e\) yields a linear automorphism of
\(\mathrm{Lie}(G)\), and the assignment \(g \mapsto \mathrm{Ad}_g\) is a smooth
homomorphism \(G \to \mathrm{GL}(\mathrm{Lie}(G))\). This homomorphism is a
Lie group representation,
the adjoint representation of \(G\).
The infinitesimal version is then the differential of \(\mathrm{Ad}\) at the
identity: \(\mathrm{ad} = d(\mathrm{Ad})_e : \mathrm{Lie}(G) \to
\mathrm{End}(\mathrm{Lie}(G))\). The theorem of this section identifies
\(\mathrm{ad}_X(Y)\) with the Lie bracket \([X, Y]\) in \(\mathrm{Lie}(G)\).
Concretely, the identification extends \(X\) and \(Y\) to left-invariant vector
fields on \(G\) and applies \(\mathcal{L}_X Y = [X, Y]\) at the identity. The flow
of a left-invariant field acts by
right translation,
while the left-invariance of \(Y\) contributes a left translation, and the two
compose into conjugation. Differentiating that family of conjugations at time
zero is what produces \(\mathrm{ad}_X\) on the left. The identification now
has explicit geometric content. The element
\(\mathrm{ad}_X(Y)\) is the rate of change of \(Y\) along the flow of \(X\). The
matrix-Lie-group instances \(\mathrm{Ad}_g(X) = g X g^{-1}\) and
\(\mathrm{ad}_X(Y) = X Y - Y X\) recover from this construction by writing the
conjugation map in matrix coordinates and computing its differential. The
bracket of left-invariant vector fields, identified with the
matrix commutator under the evaluation isomorphism,
produces the matrix expression for \(\mathrm{ad}_X(Y)\) as a consequence of the
same identification.
The Bilinear Product Rule for the Bracket, Demystified
The
bilinear product rule
for the Lie bracket, established earlier as the identity
\[
[f V, \, g W] = f g \, [V, W] + f \, (V g) \, W - g \, (W f) \, V ,
\]
was introduced as an algebraic identity whose origin in the underlying
differentiation was not made explicit. The Lie-derivative reading of the bracket
accounts for it directly. The bracket can be read in two ways. Read as a Lie
derivative in the first slot, \([f V, g W] = \mathcal{L}_{f V}(g W)\), it is the
derivative of the field \(g W\) along the flow of \(f V\), and Leibniz rule (d)
in the corollary above expands the rate of change of a product into separate
rates of change of the function-factor and the vector-field-factor. The derivative
acts on the scalar \(g\) and on the vector field \(W\) in succession. Read instead
as a Lie derivative in the second slot,
\([f V, g W] = -\mathcal{L}_{g W}(f V)\) by antisymmetry (a), the same Leibniz
rule expands the rate of change of \(f V\) along the flow of \(g W\) into
separate rates of change of \(f\) and of \(V\). The bracket is the same object
in either reading, so the two expansions are equal. Assembling the
function-derivative terms that they each contribute produces exactly the cross
terms \(f \, (V g) \, W\) and \(-g \, (W f) \, V\) of the bilinear formula.
Concretely, the first reading gives
\(\mathcal{L}_{f V}(g W) = \bigl( (f V) g \bigr) W + g \, \mathcal{L}_{f V} W
= f (V g) W + g [f V, W]\). The second reading gives
\(-\mathcal{L}_{g W}(f V) = -g (W f) V - f \, \mathcal{L}_{g W} V
= -g (W f) V + f \, [V, g W]\) after a second application of antisymmetry on
the inner Lie derivative. When these two expressions for \([f V, g W]\) are equated,
the remaining bracket terms \(g [f V, W]\) and \(f [V, g W]\) consolidate
through bilinearity of the bracket in each slot, leaving the principal term
\(f g [V, W]\) once the function-derivative cross terms have been separated
out.
The opacity of the original identity was an artifact of viewing the bracket only
algebraically. Viewed as a Lie derivative, it carries a Leibniz rule in each
slot, and the bilinear product rule is the symmetric closure of those two
rules. The algebraic identity is the combinatorial trace left by differentiation
acting once on each scalar factor.
Commuting Vector Fields and Flow Invariance
The Lie derivative of \(W\) along \(V\) measures the rate of change of \(W\) as
transported by the flow of \(V\), evaluated at \(t = 0\). The construction extends to
arbitrary times. At any time \(t_0\) for which the flow is defined, the corresponding
rate of change is expressible in terms of the Lie derivative through a single
linear transport by \(d(\theta_{-t_0})\). This is the content of the proposition
below, and it is the technical input for the equivalence between commutativity of
the bracket and invariance of one field under the flow of the other.
Proof:
Set \(q = \theta_{t_0}(p)\), and consider the change of variable \(t = t_0 + s\).
The map \(t \mapsto d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)})\) takes its
values in the single vector space \(T_p M\), and the substitution shifts the
parameter without altering the curve. Its derivative at \(t = t_0\) therefore
exists exactly when the derivative of the reparametrized curve at \(s = 0\)
exists, and the two agree:
\[
\frac{d}{dt}\bigg|_{t = t_0}
d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr)
= \frac{d}{ds}\bigg|_{s = 0}
d(\theta_{-t_0 - s})_{\theta_{t_0 + s}(p)}\bigl( W_{\theta_{t_0 + s}(p)} \bigr) .
\]
The group law of the flow applies to both arguments. The time index
\(-t_0 - s = -s + (-t_0)\) decomposes the map as \(\theta_{-t_0 - s} =
\theta_{-t_0} \circ \theta_{-s}\), and the base point satisfies
\(\theta_{t_0 + s}(p) = \theta_s(q)\). The chain rule applied to this
composition gives, at the point \(r = \theta_s(q)\),
\[
d(\theta_{-t_0 - s})_r
= d(\theta_{-t_0})_{\theta_{-s}(r)} \circ d(\theta_{-s})_r ,
\]
where \(\theta_{-s}(r) = \theta_{-s}(\theta_s(q)) = q\). Substituting back
rewrites the integrand as
\[
d(\theta_{-t_0 - s})_{\theta_{t_0 + s}(p)}\bigl( W_{\theta_{t_0 + s}(p)} \bigr)
= d(\theta_{-t_0})_q \circ d(\theta_{-s})_{\theta_s(q)}\bigl( W_{\theta_s(q)} \bigr) .
\]
The map \(d(\theta_{-t_0})_q : T_q M \to T_p M\)
is linear and independent of \(s\), so it commutes with the \(s\)-derivative:
\[
\frac{d}{ds}\bigg|_{s = 0}
d(\theta_{-t_0})_q \circ d(\theta_{-s})_{\theta_s(q)}\bigl( W_{\theta_s(q)} \bigr)
= d(\theta_{-t_0})_q \left(
\frac{d}{ds}\bigg|_{s = 0}
d(\theta_{-s})_{\theta_s(q)}\bigl( W_{\theta_s(q)} \bigr) \right) .
\]
The inner derivative is exactly the Lie derivative of \(W\) with respect to
\(V\) at the point \(q\), evaluated through the definition. Substituting yields
\(d(\theta_{-t_0})_q \bigl((\mathcal{L}_V W)_q\bigr)\), which is the right-hand
side of the claimed identity.
The proposition says, in geometric language, that the rate of change of \(W\)
transported by the flow at time \(t_0\) is the Lie derivative at \(\theta_{t_0}(p)\)
transported back to \(p\) by a single application of \(d(\theta_{-t_0})\). The Lie
derivative at \(t = 0\) controls the entire time-development of the transported
field, not just its initial value. A vanishing Lie derivative therefore forces the
transported field to be constant in time, and this is the geometric content of the
main theorem of this section.
Commuting and Flow-Invariant Vector Fields
Two vector fields whose bracket vanishes are said to commute. The terminology
reflects the fact that the action of one as a derivation commutes with the action of
the other on smooth functions. The formal definition below is just the statement
\([V, W] = 0\) in the equivalent derivation form.
Definition: Commuting Vector Fields
Two vector fields \(V, W \in \mathfrak{X}(M)\) commute if
\(V (W f) = W (V f)\) for every \(f \in C^\infty(M)\), or equivalently if
\([V, W] = 0\) as a vector field on \(M\).
A separate geometric condition expresses how a vector field can be unchanged by a
flow that may belong to another vector field altogether. The condition is the
natural one. Pushing the values of \(W\) forward by the flow of \(V\) at any time
reproduces \(W\) at the new base point.
Definition: Vector Field Invariant Under a Flow
Let \(\theta : \mathcal{D} \to M\) be a smooth flow. A vector field
\(W \in \mathfrak{X}(M)\) is said to be invariant under \(\theta\)
if
\[
d(\theta_t)_p (W_p) = W_{\theta_t(p)}
\]
for every \((t, p)\) in the flow domain. Equivalently, \(W\) is invariant under
\(\theta\) if for each \(t\) the restriction \(W|_{M_t}\) is
\(\theta_t\)-related
to \(W|_{M_{-t}}\), where \(M_t = \{p \in M : (t, p) \in \mathcal{D}\}\) is the
open set on which the time-\(t\) flow map \(\theta_t\) is defined.
The algebraic condition (commutativity of the bracket) and the geometric condition
(invariance of one field under the flow of the other) are linked by the theorem
below. Once linked symmetrically in the two fields, they are also linked to the
invariance of the first field under the flow of the second.
Theorem (Commute iff Flow-Invariant)
Let \(M\) be a smooth manifold and \(V, W \in \mathfrak{X}(M)\). The following
are equivalent.
(a) \(V\) and \(W\) commute.
(b) \(W\) is invariant under the flow of \(V\).
(c) \(V\) is invariant under the flow of \(W\).
Proof:
It suffices to prove the equivalence (a) \(\Longleftrightarrow\) (b). Applying the
same argument with the roles of \(V\) and \(W\) exchanged yields
(a) \(\Longleftrightarrow\) (c), and the equivalence of all three follows.
(b) \(\Longrightarrow\) (a).
Assume \(W\) is invariant under the flow of \(V\).
Let \(\theta\) denote the flow of \(V\). The group law of the flow gives
\(\theta_{-t} \circ \theta_t = \mathrm{id}_{M_t}\) on the open set
\(M_t = \{p \in M : (t, p) \in \mathcal{D}\}\), so the chain rule yields
\(d(\theta_{-t})_{\theta_t(p)} \circ d(\theta_t)_p = \mathrm{id}_{T_p M}\) for
every \((t, p)\) in the flow domain. That is, \(d(\theta_{-t})_{\theta_t(p)}\)
is the linear inverse of \(d(\theta_t)_p\). Applying this inverse to both
sides of the invariance identity \(d(\theta_t)_p (W_p) = W_{\theta_t(p)}\)
gives
\[
W_p = d(\theta_{-t})_{\theta_t(p)} \bigl( W_{\theta_t(p)} \bigr)
\]
for every \((t, p)\) in the flow domain. The right-hand side is the family
whose \(t\)-derivative at \(t = 0\) defines the Lie derivative. The left-hand
side is constant in \(t\). Differentiating at \(t = 0\) yields
\((\mathcal{L}_V W)_p = 0\) at every \(p\), so \(\mathcal{L}_V W \equiv 0\). The
identification of the Lie derivative with the Lie bracket established earlier
in this development gives \([V, W] = 0\), and so \(V\) and \(W\) commute.
(a) \(\Longrightarrow\) (b).
Assume \(V\) and \(W\) commute, so that
\(\mathcal{L}_V W = [V, W] = 0\) as a vector field on \(M\). Fix \(p \in M\), and
consider the curve in the vector space \(T_p M\) given by
\[
X(t) = d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr) ,
\quad t \in \mathcal{D}^{(p)} .
\]
The
pullback derivative formula
evaluates the derivative of \(X\) at any time \(t_0 \in \mathcal{D}^{(p)}\) as
\[
X'(t_0)
= d(\theta_{-t_0})_{\theta_{t_0}(p)} \bigl( (\mathcal{L}_V W)_{\theta_{t_0}(p)} \bigr) ,
\]
and the right-hand side is zero by the assumption \(\mathcal{L}_V W = 0\). The
curve \(X\) therefore has identically vanishing derivative on
\(\mathcal{D}^{(p)}\), so it is constant in \(t\). Evaluating at \(t = 0\),
the identity \(\theta_0 = \mathrm{id}_M\) of the flow at time zero gives
\(\theta_0(p) = p\) and \(d(\theta_0)_p = \mathrm{id}_{T_p M}\), so
\(X(0) = \mathrm{id}_{T_p M}(W_p) = W_p\). Hence \(X(t) = W_p\) for all
\(t \in \mathcal{D}^{(p)}\). Applying the inverse \(d(\theta_t)_p\) of
\(d(\theta_{-t})_{\theta_t(p)}\) to both sides, by the same chain-rule argument
as in the converse direction, recovers
\(W_{\theta_t(p)} = d(\theta_t)_p (W_p)\), the invariance condition at the
point \(p\). Since \(p\) was arbitrary, \(W\) is invariant under the flow of
\(V\).
A small but striking instance of the theorem applies when \(V\) and \(W\) are taken
to be the same vector field. The bracket of any vector field with itself vanishes,
so the algebraic side of the equivalence is satisfied trivially. The geometric side
therefore holds as well.
Corollary (Self-Invariance Under Own Flow)
Every smooth vector field is invariant under its own flow.
Proof:
For any \(V \in \mathfrak{X}(M)\), the bracket \([V, V]\) vanishes by
antisymmetry. Applying the theorem above with \(W = V\), in the form of the
equivalence (a) \(\Longleftrightarrow\) (b), gives the invariance of \(V\) under
its own flow.
The corollary recovers, at a higher level of abstraction, a fact already implicit in
the construction of integral curves. The value of \(V\) at the point reached after
flowing for time \(t\) is the pushforward of its value at the starting point by the
flow itself. The geometric reading is that integral curves are everywhere tangent to
the field they generate, and the algebraic reading is that the field commutes with
itself. The two readings are formally the same equivalence applied in the
degenerate case \(V = W\).
Commuting Flows
The deepest characterization of commuting vector fields is in terms of the
relationship between their respective flows. The theorem of this section asserts
that two smooth vector fields commute precisely when their flows commute as maps,
not merely as infinitesimal generators. Before stating the result, we have to
decide what it should mean for two flows to commute when neither of the vector
fields is complete. The naive notion of commutativity needs a small adjustment to
account for the limited domains of partial flows.
What It Means for Flows to Commute
For two
complete
vector fields \(V\) and \(W\) with global flows \(\theta\) and \(\psi\) on \(M\), the
commutation of the flows means simply
\[
\theta_t \circ \psi_s = \psi_s \circ \theta_t
\quad \text{for all } s, t \in \mathbb{R} .
\]
Both sides are defined on all of \(M\), and the assertion is the equality of two
diffeomorphisms.
When either field fails to be complete, this naive formulation cannot hold as
written, because each of \(\theta_t \circ \psi_s(p)\) and \(\psi_s \circ \theta_t(p)\)
is defined only when a sequence of partial flows can be composed without leaving
their respective flow domains. The most that one can reasonably require is the
equation \(\theta_t \circ \psi_s(p) = \psi_s \circ \theta_t(p)\) at points where both
sides are defined. A subtlety appears at exactly this point. There are
examples of commuting vector fields and pairs \((t, s, p)\) for which both
\(\theta_t \circ \psi_s(p)\) and \(\psi_s \circ \theta_t(p)\) happen to be defined and
yet are unequal. The obstruction is that the value at one pair \((t, s)\) does not
extend along intermediate values. If \(\theta_t \circ \psi_s(p)\) is defined at
\((t_0, s_0)\), then by the openness of the flow domain it is defined for all \(t\)
in an open interval containing \(0\) and \(t_0\). The analogous statement need not
hold along the \(s\)-axis, because the integral curve of \(V\) starting at
\(\psi_s(p)\) may fail to extend to time \(t_0\) for some intermediate \(s\) between
\(0\) and \(s_0\). Equality at a single \((t_0, s_0)\) is therefore the wrong
requirement. What is wanted is equality on a rectangle along which both sides admit
a continuous deformation back to the identity.
Definition: Commuting Flows
Let \(\theta\) and \(\psi\) be smooth flows on a smooth manifold \(M\). The
flows \(\theta\) and \(\psi\) commute if the following condition
holds at every \(p \in M\). Whenever \(J\) and \(K\) are open intervals containing
\(0\) such that one of the two expressions \(\theta_t \circ \psi_s(p)\) or
\(\psi_s \circ \theta_t(p)\) is defined for every \((s, t) \in J \times K\),
both are defined on \(J \times K\) and are equal there.
For global flows, this condition reduces to the assertion that
\(\theta_t \circ \psi_s = \psi_s \circ \theta_t\) as diffeomorphisms of \(M\) for
all \(s, t \in \mathbb{R}\).
The Theorem
Theorem (Commute iff Flows Commute)
Smooth vector fields \(V\) and \(W\) on a smooth manifold \(M\) commute if and
only if their flows commute in the sense of the definition above.
Proof:
Let \(\theta\) and \(\psi\) denote the flows of \(V\) and \(W\), respectively.
Vector fields commute \(\Longrightarrow\) flows commute.
Assume \([V, W] = 0\), fix \(p \in M\), and let \(J, K\) be open intervals
containing \(0\) such that \(\psi_s \circ \theta_t(p)\) is defined for every
\((s, t) \in J \times K\). The argument with the roles of the two flows exchanged
covers the analogous case for \(\theta_t \circ \psi_s(p)\). The
equivalence of commuting and flow-invariance
applied in the form (c) of that theorem gives that \(V\) is invariant under the
flow of \(W\), so for every \(s \in J\) and every \(q\) in the domain of
\(\psi_s\),
\[
d(\psi_s)_q (V_q) = V_{\psi_s(q)} .
\]
Fix \(s \in J\) and consider the curve \(\gamma : K \to M\) defined by
\[
\gamma(t) = \psi_s \circ \theta_t(p) = \psi_s \bigl( \theta^{(p)}(t) \bigr) .
\]
Its starting point is \(\gamma(0) = \psi_s(p)\), and its velocity at any
\(t \in K\) is the differential of \(\psi_s\) applied to the velocity of
\(\theta^{(p)}\):
\[
\begin{align*}
\gamma'(t)
&= d(\psi_s)_{\theta^{(p)}(t)}\bigl( \theta^{(p)\prime}(t) \bigr) \\\\
&= d(\psi_s)_{\theta^{(p)}(t)}\bigl( V_{\theta^{(p)}(t)} \bigr) \\\\
&= V_{\psi_s(\theta^{(p)}(t))} \\\\
&= V_{\gamma(t)},
\end{align*}
\]
the middle equality using the integral-curve equation
\(\theta^{(p)\prime}(t) = V_{\theta^{(p)}(t)}\), and the penultimate equality the
invariance of \(V\) under \(\psi_s\) applied at the point
\(\theta^{(p)}(t)\). So \(\gamma\) is an integral curve of \(V\) starting at
\(\psi_s(p)\). The
absorption of integral curves by trajectories
therefore places \(K\) inside the maximal interval
\(\mathcal{D}^{(\psi_s(p))}\) and identifies \(\gamma\) with the restriction of
the trajectory \(\theta^{(\psi_s(p))}\) to \(K\). Hence for every \(t \in K\),
\[
\gamma(t) = \theta^{(\psi_s(p))}(t) = \theta_t \circ \psi_s(p) .
\]
Combining the two expressions for \(\gamma(t)\) yields
\(\psi_s \circ \theta_t(p) = \theta_t \circ \psi_s(p)\) for every
\((s, t) \in J \times K\), which is the commutation of the flows at \(p\). Since
\(p\) was arbitrary, the flows commute.
Flows commute \(\Longrightarrow\) vector fields commute.
Assume the flows commute. Fix \(p \in M\) and choose \(\varepsilon \gt 0\) small enough that
\(\psi_s \circ \theta_t(p)\) is defined for every \(|s|, |t| \lt \varepsilon\).
The hypothesis gives
\(\psi_s \circ \theta_t(p) = \theta_t \circ \psi_s(p)\) for all such \(s, t\). In
terms of the maximal integral curves of \(W\), this reads
\[
\psi^{(\theta_t(p))}(s) = \theta_t\bigl( \psi^{(p)}(s) \bigr) .
\]
Both sides are smooth curves in \(s\) starting at \(\theta_t(p)\) when
\(s = 0\). Differentiating the left side at \(s = 0\) yields the value
\(W_{\theta_t(p)}\) of the integral curve of \(W\) at its starting point, and
differentiating the right side yields the differential \(d(\theta_t)_p\) applied to
the velocity \(\psi^{(p)\prime}(0) = W_p\). Equating gives
\[
W_{\theta_t(p)} = d(\theta_t)_p (W_p)
\quad \text{for every } |t| \lt \varepsilon .
\]
The map \(d(\theta_{-t})_{\theta_t(p)}\) is the inverse of \(d(\theta_t)_p\) on
tangent spaces, by the same chain-rule argument used earlier. Applying it to both
sides produces
\[
d(\theta_{-t})_{\theta_t(p)}\bigl( W_{\theta_t(p)} \bigr) = W_p
\]
for every \(|t| \lt \varepsilon\). The right-hand side is the constant vector
\(W_p\), independent of \(t\). The left-hand side is therefore the constant
function \(t \mapsto W_p\) as a curve in \(T_p M\). Its \(t\)-derivative at
\(t = 0\) is consequently zero, and by the definition of the Lie derivative
this derivative equals \((\mathcal{L}_V W)_p\), so \((\mathcal{L}_V W)_p = 0\).
Since \(p\) was arbitrary, \(\mathcal{L}_V W \equiv 0\) on \(M\), and the
identification of the Lie derivative with the Lie bracket gives \([V, W] = 0\).
The vector fields commute.
The theorem completes a four-step chain of equivalent conditions on a pair of
smooth vector fields: bracket vanishes, Lie derivative vanishes, each field is
invariant under the flow of the other, and the two flows commute as maps. Each step
of the chain has its own flavor, whether algebraic, infinitesimal-geometric, or
finite-geometric. The equivalence among them is the content of the
bracket-flow correspondence at the level of pairs of vector fields. The
\(k\)-tuple generalization is the subject of the next section.
Commuting Frames and Canonical Coordinates
The
canonical form theorem near a regular point
showed that a single nonvanishing smooth vector field can be straightened locally
into the coordinate vector field \(\partial / \partial s^1\). The natural
generalization straightens a whole \(k\)-tuple of vector fields simultaneously onto
a \(k\)-tuple of coordinate vector fields \(\partial / \partial s^i\), and it is not
automatic. An arbitrary \(k\)-tuple of nowhere-vanishing fields need not be
realizable as a coordinate frame, and the obstruction is exactly the commutativity
of the brackets. The structural identity of the previous sections, that the bracket
vanishes if and only if the flows commute, turns out to be the precise condition
under which the simultaneous straightening can be carried out.
Local Frames and Commuting Frames
Definition: Local Frame
Let \(M\) be a smooth \(n\)-manifold. A local frame for \(M\)
is an \(n\)-tuple \((E_1, \ldots, E_n)\) of vector fields defined on an open
subset \(U \subseteq M\) such that \((E_1|_p, \ldots, E_n|_p)\) is a basis of
\(T_p M\) at every \(p \in U\). A local frame is smooth if each
\(E_i\) is a smooth vector field on \(U\).
Definition: Commuting Frame
A smooth local frame \((E_1, \ldots, E_n)\) for \(M\) is called a
commuting frame if \([E_i, E_j] = 0\) for all \(i, j\). Some
authors call commuting frames holonomic frames.
The simplest example of a commuting frame is the coordinate frame attached to any
chart. Given a smooth chart \((U, (x^i))\), the coordinate vector fields
\(\partial / \partial x^i\) on \(U\) form a smooth local frame, and the
coordinate-frame bracket identity
asserts \([\partial / \partial x^i, \, \partial / \partial x^j] = 0\) for all
\(i, j\). The frame is therefore commuting. The same statement read
contrapositively gives a necessary condition. A smooth frame with a nonvanishing
bracket at some point cannot agree with the coordinate frame of a chart on any
neighborhood of that point, since the brackets of a coordinate frame vanish
identically on the chart domain. The theorem below settles whether the
commuting condition is also
sufficient, that is, whether every smooth commuting frame is locally a
coordinate frame.
Commutativity is not automatic for arbitrary smooth frames. As a concrete
illustration on \(\mathbb{R}^2 \setminus \{0\}\), consider the
polar
orthonormal frame
obtained by orthonormalizing the radial and angular vector fields. Its bracket
computes to
\[
[E_1, E_2]
= \frac{y}{r^2} \, \frac{\partial}{\partial x}
- \frac{x}{r^2} \, \frac{\partial}{\partial y}
\neq 0 ,
\]
so the polar orthonormal frame is not a commuting frame. By the necessity direction
just established, the frame cannot be the coordinate frame of any chart on
\(\mathbb{R}^2 \setminus \{0\}\).
The Canonical Form Theorem for Commuting Vector Fields
Proof:
If no submanifold \(S\) is supplied, construct one. Since \((V_1|_p, \ldots, V_k|_p)\) is linearly independent
in \(T_p M\), it can be extended to a basis of \(T_p M\). Choose any smooth chart \((U, (x^i))\) centered at
\(p\) with respect to which the chosen extension is
\[
(V_1|_p, \ldots, V_k|_p, \partial / \partial x^{k+1}|_p, \ldots, \partial / \partial x^n|_p).
\]
Take \(S\) to be the slice \(\{x^1 = \cdots = x^k = 0\}\) in this chart. The tangent space of \(S\)
at \(p\) is the span of
\[
(\partial / \partial x^{k+1}|_p, \ldots, \partial / \partial x^n|_p),
\]
which is complementary to the span of \((V_1|_p, \ldots, V_k|_p)\) by the choice
of basis. If \(S\) was supplied, use instead a slice chart
\((U, (x^i))\) for \(S\) centered at \(p\), which the
local slice criterion
provides because \(S\) is embedded, and which realizes \(S \cap U\) as
\(\{x^1 = \cdots = x^k = 0\}\). Either way, parametrize a neighborhood of
\(p\) in \(S\) by an open subset
\(\Omega \subseteq \mathbb{R}^{n - k}\) with coordinates
\((s^{k+1}, \ldots, s^n)\) so that \(S\) corresponds to
\(\{(0, \ldots, 0, s^{k+1}, \ldots, s^n) : (s^{k+1}, \ldots, s^n) \in \Omega\}\)
in the chart, with \(s^{k+1}|_p = \cdots = s^n|_p = 0\).
Let \(\theta_i\) denote the flow of \(V_i\) for \(i = 1, \ldots, k\). The
openness of each flow domain and the smoothness of compositions of partial flows
yield an \(\varepsilon \gt 0\) and an open neighborhood \(Y \subseteq U\) of \(p\)
such that the composition
\[
(\theta_1)_{s^1} \circ (\theta_2)_{s^2} \circ \cdots \circ (\theta_k)_{s^k}
\]
is defined on \(Y\) and takes values in \(U\) whenever
\(|s^1|, \ldots, |s^k| \lt \varepsilon\). Shrinking \(\Omega\) if necessary, define
the map
\[
\Phi : (-\varepsilon, \varepsilon)^k \times \Omega \to U ,
\]
\[
\Phi(s^1, \ldots, s^k, s^{k+1}, \ldots, s^n)
= (\theta_1)_{s^1} \circ \cdots \circ (\theta_k)_{s^k}
(0, \ldots, 0, s^{k+1}, \ldots, s^n) .
\]
Setting all of \(s^1, \ldots, s^k\) equal to zero collapses each partial flow
\((\theta_i)_0\) to the identity map, since the flow of any smooth vector
field is the identity at time zero:
\[
\Phi(0, \ldots, 0, s^{k+1}, \ldots, s^n) =
(0, \ldots, 0, s^{k+1}, \ldots, s^n).
\]
Hence \(\Phi(\{0\} \times \Omega)\) is exactly the image of \(\Omega\) under the chart
parametrization of \(S\). In particular, \(\Phi\) sends the slice
\(\{s^1 = \cdots = s^k = 0\}\) in its domain onto a neighborhood of \(p\) in \(S\).
We show next that \(\Phi\) sends each coordinate vector field
\(\partial / \partial s^i\) for \(i = 1, \ldots, k\) to \(V_i\). Fix
\(i \in \{1, \ldots, k\}\) and a point \(s_0 = (s_0^1, \ldots, s_0^n)\) in the
domain of \(\Phi\). The flows of any two of \(V_1, \ldots, V_k\) commute, a consequence of the
commute-iff-flows-commute theorem
applied to each pair \((V_i, V_j)\). The composition defining \(\Phi\) can therefore
be rewritten so that \((\theta_i)_{s^i}\) appears in the leftmost position:
\[
\Phi(s)
= (\theta_i)_{s^i} \,\circ\, \Bigl[ (\theta_1)_{s^1} \circ \cdots \circ
\widehat{(\theta_i)_{s^i}} \circ \cdots \circ (\theta_k)_{s^k} \Bigr]
(0, \ldots, 0, s^{k+1}, \ldots, s^n) ,
\]
where the hat denotes omission of the \(i\)-th factor. The bracketed inner
expression is independent of \(s^i\). Writing it as a function of the remaining
coordinates, call its value at \(s = s_0\) the point
\(q = q(s_0^1, \ldots, \widehat{s_0^i}, \ldots, s_0^n)\). The
\(s^i\)-derivative at \(s_0\) is then the derivative of the single-flow curve
\(t \mapsto (\theta_i)_t(q)\) at \(t = s_0^i\). This is the velocity of the
integral curve of \(V_i\) through \(q\) at parameter \(s_0^i\), which equals
\(V_i\) evaluated at \((\theta_i)_{s_0^i}(q) = \Phi(s_0)\). Hence
\[
d\Phi_{s_0}\!\left( \frac{\partial}{\partial s^i}\bigg|_{s_0} \right)
= V_i\!\big|_{\Phi(s_0)}
\quad \text{for } i = 1, \ldots, k .
\]
For \(i = k + 1, \ldots, n\) the analogous identity follows from
\[
\Phi(0, \ldots, 0, s^{k+1}, \ldots, s^n) = (0, \ldots, 0, s^{k+1}, \ldots, s^n),
\]
evaluated at \(s_0 = 0\). Differentiating gives
\[
d\Phi_0(\partial / \partial s^i|_0) = \partial / \partial x^i|_p.
\]
At the origin \(s_0 = 0\), the differential \(d\Phi_0\) therefore maps the basis
\((\partial / \partial s^1|_0, \ldots, \partial / \partial s^n|_0)\) of
\(T_0 \mathbb{R}^n\) to the tuple
\[
\bigl(V_1|_p, \ldots, V_k|_p, \, \partial / \partial x^{k+1}|_p, \ldots, \partial / \partial x^n|_p\bigr)
\]
in \(T_p M\). The complementarity hypothesis guarantees that this tuple is a basis of \(T_p M\), so
\(d\Phi_0\) is a linear isomorphism. By the inverse function theorem, \(\Phi\) is a diffeomorphism on
some open neighborhood of \(0\) onto its image, which is an open neighborhood of
\(p\) in \(M\). Setting \(\varphi = \Phi^{-1}\), the coordinates
\((s^1, \ldots, s^n) = \varphi\) form a smooth chart centered at \(p\). The
pushforward relation just established gives \(\varphi_* V_i = \partial /
\partial s^i\) for \(i = 1, \ldots, k\), and the construction ensures that \(S\)
corresponds to the slice \(\{s^1 = \cdots = s^k = 0\}\) in these coordinates.
The proof of the theorem furnishes a construction in addition to an existence
statement. To straighten a commuting tuple \((V_1, \ldots, V_k)\) into
coordinate vector fields, one chooses an \((n - k)\)-dimensional transverse
submanifold \(S\) through the point of interest, parametrizes it by an open subset
of \(\mathbb{R}^{n - k}\), and follows the \(k\) flows
\((\theta_1)_{s^1}, \ldots, (\theta_k)_{s^k}\) in any order. Commutativity of the
flows ensures the order is immaterial. The inverse of the resulting parametrization
\(\Phi\) is the desired chart, and \(V_i\) becomes \(\partial / \partial s^i\) by
construction.
Polar and Logarithmic-Polar Coordinates from a Commuting Pair
On \(\mathbb{R}^2 \setminus \{0\}\), the rotation and dilation vector fields
\[
V = x \, \frac{\partial}{\partial y} - y \, \frac{\partial}{\partial x} ,
\quad
W = x \, \frac{\partial}{\partial x} + y \, \frac{\partial}{\partial y}
\]
have vanishing bracket, as a direct computation gives \([V, W] = 0\), and form a
linearly independent pair at every point of \(\mathbb{R}^2 \setminus \{0\}\). Their
flows can be written down by inspection. The rotation generator \(V\) generates the
angular flow
\(\theta_t(x, y) = (x \cos t - y \sin t, \, x \sin t + y \cos t)\), and the dilation
generator \(W\) generates the radial flow
\(\eta_t(x, y) = (e^t x, e^t y)\).
The canonical-form construction applied to \((V, W)\) at the point \(p = (1, 0)\)
proceeds as follows. Since \(V|_p = \partial / \partial y|_{(1,0)}\) and
\(W|_p = \partial / \partial x|_{(1,0)}\) together span all of
\(T_{(1,0)} \mathbb{R}^2\), the submanifold \(S\) can be taken to be the single
point \(\{(1, 0)\}\). Here \(n = 2\), so that point has codimension \(k = 2\), as
the theorem requires. The composite map
\(\Phi : \mathbb{R}^2 \to \mathbb{R}^2 \setminus \{0\}\) is
\[
\Phi(s, t)
= \eta_t \circ \theta_s (1, 0)
= \eta_t (\cos s, \sin s)
= (e^t \cos s, \, e^t \sin s) .
\]
Inverting locally near \((1, 0)\), the coordinate map is
\[
(s, t) = \Phi^{-1}(x, y)
= \left( \tan^{-1}(y / x), \, \log\!\sqrt{x^2 + y^2} \right) .
\]
In these coordinates, \(V = \partial / \partial s\) and \(W = \partial / \partial t\)
by the theorem.
The pair \((s, t)\) is the system of logarithmic polar coordinates. The
first coordinate is the angular coordinate already encountered in the canonical
form of the rotation field alone, and the second is the logarithm of the radial
coordinate. The shift from radial coordinate to its logarithm is the geometric
signature of having included \(W\) in the commuting pair. The radial dilation flow
acts by multiplication on the radial coordinate, so flattening that action to
translation in a coordinate variable requires the logarithm of the radial distance
as the natural variable. The same geometric principle that produced ordinary polar
coordinates as the canonical form of the rotation field alone yields logarithmic
polar coordinates as the canonical form of the rotation-dilation pair.
The Bracket-Flow Correspondence
The four theorems of this development together constitute the bracket-flow
correspondence for smooth vector fields on a manifold. They are the identification
of the Lie derivative with the bracket, the equivalence of bracket-commutativity
with flow-invariance, the equivalence of vector-field commutativity with flow
commutativity, and the canonical form for a commuting frame. The correspondence
pairs algebraic data with geometric data at four levels of increasing strength:
| Algebraic side |
Geometric side |
| The bracket \([V, W]\) |
The rate of change of \(W\) along the flow of \(V\) |
| \([V, W] = 0\) |
Each field is invariant under the flow of the other |
| \([V, W] = 0\) |
The flows of \(V\) and \(W\) commute as maps |
| A \(k\)-tuple with all pairwise brackets zero |
Coordinates in which \(V_i = \partial / \partial s^i\) |
Two large bodies of application read the correspondence in opposite directions.
In geometric deep learning, the symmetries acting on data are typically encoded
as a Lie group, and the
left-invariant vector fields
on that group are the infinitesimal generators of one-parameter symmetry
subgroups. An abelian subalgebra of the
Lie algebra,
for instance the algebra of a maximal torus, corresponds exactly to a tuple of
commuting vector fields. The canonical form for commuting fields then produces
coordinates in which the torus action is translation, the local model underlying
equivariant network constructions on those groups. In control
theory, an admissible control system on a manifold prescribes a set of input
vector fields, and the reachable set under independent control of those fields
is parametrized by the very composite map \(\Phi\) of the canonical-form
construction when the fields commute. A nonzero bracket \([g_i, g_j]\) marks the
breakdown of commutativity, and it generates new infinitesimal directions of
reachability, the algebraic seed of the controllability theory of nonholonomic
systems.