The Differential and Vector Fields
Once vector fields are at hand, two natural operations arise: pushing them along smooth maps between
manifolds, and combining them with one another. The first operation, governed by the differential,
leads to a refined notion of compatibility called \(F\)-relatedness, and clarifies when a vector
field on one manifold transfers cleanly to another. The second operation, the Lie bracket
\([X, Y] = XY - YX\), produces a new vector field from two given ones and turns
\(\mathfrak{X}(M)\)
into an algebraic object whose structure mirrors the matrix commutator structure already used in the
linear-algebra track. The bracket measures the failure of two flows to commute, and its algebraic
structure carries the theory of vector fields toward Lie algebras.
The differential does not move vector fields cleanly
Suppose \(F : M \to N\) is smooth and
\(X\) is a vector field
on \(M\). For each \(p \in M\), the
differential
\(dF_p : T_pM \to T_{F(p)}N\) sends the tangent vector \(X_p\) to a tangent vector
\(dF_p(X_p) \in T_{F(p)}N\). The question is whether the assignment \(q \mapsto dF_p(X_p)\) for points
\(q \in N\) of the form \(q = F(p)\) extends to a well-defined vector field on \(N\). In general it
does not.
Two obstructions arise. If \(F\) is not surjective, there are points \(q \in N \setminus F(M)\) at
which no rule has been specified, and one cannot decide what tangent vector to assign there. If
\(F\) is not injective, there are points \(q \in N\) with several preimages \(p_1, p_2, \dots\), and
the differential will in general send the corresponding tangent vectors \(X_{p_1}, X_{p_2}, \dots\)
to different tangent vectors at \(q\), and there is no preferred choice among them. The differential
is a pointwise object, indifferent to the global question of whether the values it produces fit
together into a section.
For the construction to succeed we need either a hypothesis on \(F\) that rules out both obstructions,
or a hypothesis relating \(X\) to a candidate field \(Y \in \mathfrak{X}(N)\) that says the
obstructions happen not to occur for this particular pair. The latter is the more flexible notion
and arises first.
\(F\)-related vector fields
The definition asks that the value of \(Y\) at any image point \(F(p)\) coincide with what the
differential of \(F\) does to \(X_p\). It says nothing about what \(Y\) does at points of \(N\)
outside the image of \(F\), and it is compatible with multiple preimages only when the differential
sends every preimage's \(X\)-value to the same tangent vector. For a general \(F\) and a given \(X\),
there need not be any \(Y\) that is \(F\)-related to \(X\), and when there is one, it need not be
unique. The diffeomorphism case (next section) is precisely the case in which existence and
uniqueness are automatic.
The relation has a clean reformulation in terms of how the two vector fields act on smooth
functions. On the \(M\)-side, a vector field \(X\) acts on the composite \(f \circ F\), where
\(f \in C^\infty(N)\). On the \(N\)-side, the vector field \(Y\) acts on \(f\) and the result is
pulled back along \(F\). The two operations agree exactly when \(X\) and \(Y\) are \(F\)-related.
Proof:
For \(p \in F^{-1}(V)\) and \(f\) defined in a neighborhood of \(F(p)\), the
defining property of the differential
gives
\[
X_p (f \circ F) = dF_p(X_p)\, f ,
\]
while the action of \(Y\) on \(f\) at the image point yields
\[
(Yf)(F(p)) = Y_{F(p)}\, f .
\]
Both identities hold for every \(p\) and every \(f\). Pointwise, the two expressions
\(X_p(f \circ F)\) and \((Yf)(F(p))\) are equal at \(p\) if and only if
\(dF_p(X_p)\, f = Y_{F(p)}\, f\). Quantifying over all \(f\) in a
neighborhood of \(F(p)\), they are equal at \(p\) for every such \(f\) if and only if
\(dF_p(X_p) = Y_{F(p)}\). The function-level equation \(X(f \circ F) = (Yf) \circ F\) on
\(F^{-1}(V)\) for every \(V\) and every \(f\) is therefore equivalent to the pointwise tangent
vector equation \(dF_p(X_p) = Y_{F(p)}\) for every \(p\), which is precisely \(F\)-relatedness.
The function-level reformulation is the principal tool of the section. Most identities involving
\(F\)-related fields will be proved by chasing functions through both sides, since the algebraic
side is easier to manipulate than the pointwise side. We will use this proposition repeatedly when
establishing the naturality of the Lie bracket later in this page.
Example (A Simple \(F\)-Relatedness):
Let \(F : \mathbb{R} \to \mathbb{R}^2\) be the smooth map \(F(t) = (\cos t, \sin t)\), the standard
parametrization of the unit circle. The vector field \(d/dt \in \mathfrak{X}(\mathbb{R})\) is
\(F\)-related to the rotational vector field
\[
Y = -y\, \frac{\partial}{\partial x} + x\, \frac{\partial}{\partial y}
\in \mathfrak{X}(\mathbb{R}^2) .
\]
To verify this from the definition, note that the differential of \(F\) at \(t\) sends \(d/dt\)
to \(F'(t) = (-\sin t, \cos t) \in T_{F(t)}\mathbb{R}^2\), while evaluating \(Y\) at \(F(t)\)
gives \(Y_{F(t)} = -\sin t\, \partial/\partial x + \cos t\, \partial/\partial y\). These are the
same tangent vector under the canonical identification of \(T_q \mathbb{R}^2\) with
\(\mathbb{R}^2\). The \(F\)-relatedness can equivalently be checked through the action on
functions using the proposition above.
Pushforward by a Diffeomorphism
The two obstructions to transporting a vector field along a smooth map, non-surjectivity and
non-injectivity, both vanish exactly when the map is a
diffeomorphism.
In that case the resulting vector field exists, is uniquely determined, and is automatically smooth.
Of all the compatibility scenarios for transporting vector fields, this one delivers the cleanest
guarantees, and it is the case in which the differential actually moves vector fields cleanly from
one manifold to another.
Existence and uniqueness
Proposition (Pushforward Along a Diffeomorphism)
Let \(M\) and \(N\) be smooth manifolds with or without boundary, and let \(F : M \to N\) be a
diffeomorphism. For every \(X \in \mathfrak{X}(M)\) there is a unique smooth vector field
\(Y \in \mathfrak{X}(N)\) that is \(F\)-related to \(X\), given by the formula
\[
Y_q = dF_{F^{-1}(q)}\bigl( X_{F^{-1}(q)} \bigr) \quad \text{for } q \in N .
\]
Proof:
Uniqueness.
If \(Y \in \mathfrak{X}(N)\) is \(F\)-related to \(X\), then by definition
\(dF_p(X_p) = Y_{F(p)}\) for every \(p \in M\). Substituting \(p = F^{-1}(q)\), which is
well-defined because \(F\) is a bijection, gives \(Y_q = dF_{F^{-1}(q)}(X_{F^{-1}(q)})\).
Uniqueness is automatic from this formula.
Existence and smoothness.
Define \(Y : N \to TN\) by the formula above.
Since \(X_{F^{-1}(q)} \in T_{F^{-1}(q)}M\) and the differential
\(dF_{F^{-1}(q)} : T_{F^{-1}(q)}M \to T_q N\) sends tangent vectors at \(F^{-1}(q)\) to tangent
vectors at \(F(F^{-1}(q)) = q\), we have \(Y_q \in T_q N\), so \(Y\) is a rough vector field on
\(N\). It is \(F\)-related to \(X\) by construction, since substituting \(q = F(p)\) into the
formula yields \(Y_{F(p)} = dF_p(X_p)\). It remains to check that \(Y\) is smooth.
Writing the formula as the composition
\[
N \xrightarrow{F^{-1}} M \xrightarrow{X} TM \xrightarrow{dF} TN ,
\]
each factor is smooth: \(F^{-1}\) because \(F\) is a diffeomorphism, \(X\) because it is a smooth
vector field, and the
global differential
\(dF : TM \to TN\) because the differential of a smooth map is itself smooth. The composition is
therefore smooth, and \(Y\) is a smooth vector field on \(N\).
The pushforward of a vector field
The unique \(F\)-related vector field guaranteed by the previous proposition is given a name and a
notation, and one warning about the notation is in order before the definition.
Definition: Pushforward of a Vector Field
Let \(F : M \to N\) be a diffeomorphism and \(X \in \mathfrak{X}(M)\). The pushforward of
\(X\) by \(F\), denoted \(F_* X\), is the unique smooth vector field on \(N\) that is
\(F\)-related to \(X\). It is given explicitly by
\[
(F_* X)_q = dF_{F^{-1}(q)}\bigl( X_{F^{-1}(q)} \bigr) \quad \text{for } q \in N .
\]
The Symbol \(F_*\) and the Symbol \(dF\)
Two related operations associated with a smooth map \(F : M \to N\) move tangent data forward,
and the literature uses overlapping notation for them. We fix the convention used here.
At a single point \(p \in M\), the
differential of \(F\)
is the linear map \(dF_p : T_pM \to T_{F(p)}N\), and the
global differential
\(dF : TM \to TN\) assembles the pointwise differentials into a smooth map between tangent
bundles. Both require only that \(F\) be smooth, and both are denoted \(dF\) (with or without a
base-point subscript) throughout this development. Some textbooks write \(F_*\) for these
operations as well. We do not, in order to keep the next item unambiguous.
On a vector field \(X \in \mathfrak{X}(M)\), the pushforward just defined produces another
vector field \(F_* X \in \mathfrak{X}(N)\). This is a global operation, defined only when \(F\)
is a diffeomorphism, because both injectivity and surjectivity are needed for the formula to
make sense and for the result to be a vector field rather than a partial or multivalued
assignment. The pointwise identity \((F_* X)_{F(p)} = dF_p(X_p)\) is just \(F\)-relatedness
applied to \(Y = F_* X\), and it is the only place where the two notations meet in our usage.
The symbol \(dF\) acts on tangent vectors at a point, while \(F_*\) acts on global vector fields
under a diffeomorphism.
A concrete computation
When \(F^{-1}\) is given by an explicit formula, the definition above is a calculation rule.
Substitute \(F^{-1}(q)\) into the components of \(X\), then apply the Jacobian matrix of \(F\) at
\(F^{-1}(q)\). The following example carries this out for two open submanifolds of \(\mathbb{R}^2\).
Example (Computing a Pushforward in the Plane):
Let
\[
\begin{align*}
M &= \{ (x, y) : y \gt 0,\ x + y \gt 0 \} , \\\\
N &= \{ (u, v) : u \gt 0,\ v \gt 0 \} ,
\end{align*}
\]
regarded as open submanifolds of \(\mathbb{R}^2\), and define
\(F : M \to N\) by \(F(x, y) = (x + y,\ x/y + 1)\). Solving \(u = x + y\) and \(v = x/y + 1\) for
\((x, y)\) gives the inverse \(F^{-1}(u, v) = (u - u/v,\ u/v)\), and direct substitution
confirms that \(F\) is a diffeomorphism. Consider the smooth vector field
\[
X = y^2\, \frac{\partial}{\partial x}
\in \mathfrak{X}(M) .
\]
The Jacobian matrix of \(F\) at a point \((x, y) \in M\) is
\[
DF(x, y)
= \begin{pmatrix} 1 & 1 \\\\ 1/y & -x/y^2 \end{pmatrix} ,
\]
and evaluating at \((x, y) = F^{-1}(u, v) = (u - u/v,\ u/v)\) gives
\[
DF\bigl( F^{-1}(u, v) \bigr)
= \begin{pmatrix} 1 & 1 \\\\ v/u & (v - v^2)/u \end{pmatrix} ,
\]
where the lower-right entry is \(-x/y^2 = -\,(u - u/v)/(u/v)^2 = (v - v^2)/u\) after
simplification.
On the side of \(X\), evaluating at \(F^{-1}(u, v)\) gives
\[
\begin{align*}
X_{F^{-1}(u, v)}
&= (u/v)^2\, \frac{\partial}{\partial x}\bigg|_{F^{-1}(u, v)} \\\\
&= \frac{u^2}{v^2}\, \frac{\partial}{\partial x}\bigg|_{F^{-1}(u, v)} .
\end{align*}
\]
Applying the Jacobian \(DF(F^{-1}(u, v))\) to the coordinate column
\((u^2/v^2,\ 0)^\top\) representing this tangent vector, the first row produces \(u^2/v^2\) and
the second row produces \((v/u) \cdot (u^2/v^2) = u/v\). By the explicit formula for the
pushforward,
\[
(F_* X)_{(u, v)}
= \frac{u^2}{v^2}\, \frac{\partial}{\partial u}\bigg|_{(u, v)}
+ \frac{u}{v}\, \frac{\partial}{\partial v}\bigg|_{(u, v)} .
\]
As a coordinate-free identity between vector fields on \(N\),
\[
F_* X = \frac{u^2}{v^2}\, \frac{\partial}{\partial u} + \frac{u}{v}\, \frac{\partial}{\partial v} .
\]
The action of the pushforward on functions
Applying the function-level characterization of \(F\)-relatedness from the previous section to the
pair \((X, F_* X)\) yields a clean identity relating the actions of \(X\) and \(F_* X\) on smooth
functions. It is the operator-level statement that pushing the vector field forward by \(F\)
corresponds to changing the variable from a function on \(N\) to a function on \(M\) via composition
with \(F\).
Corollary (Action of the Pushforward on Functions)
Let \(F : M \to N\) be a diffeomorphism, \(X \in \mathfrak{X}(M)\), and \(f \in C^\infty(N)\).
Then
\[
\bigl( (F_* X) f \bigr) \circ F = X (f \circ F) .
\]
Proof:
By definition, \(F_* X\) is the unique vector field on \(N\) that is \(F\)-related to \(X\).
Applying the
function-level characterization of \(F\)-relatedness
with \(Y = F_* X\) gives \(X(f \circ F) = ((F_* X) f) \circ F\) on \(F^{-1}(N) = M\), which is
the stated identity.
Vector Fields and Submanifolds
Submanifolds raise a question parallel to the one just settled for smooth maps. When does a vector
field on the ambient manifold \(M\) restrict to a vector field on a submanifold \(S \subseteq M\)?
The pointwise obstruction is that the value \(X_p\) of an ambient vector field at a point of \(S\)
need not lie in the tangent space \(T_pS\). When it does at every point, the field is said to be
tangent to \(S\), and the restriction goes through. The criterion that detects tangency through the
action of the field on smooth functions specializes the
tangent-space-as-annihilator
characterization of \(T_pS\) to a global statement about vector fields.
Tangency to a submanifold
Definition: Tangent to a Submanifold
Let \(M\) be a smooth manifold, \(S \subseteq M\) an
immersed
or
embedded submanifold,
and \(X \in \mathfrak{X}(M)\). The vector field \(X\) is tangent to \(S\) at
\(p\) for \(p \in S\) if \(X_p \in T_pS \subseteq T_pM\). It is tangent to
\(S\) if it is tangent to \(S\) at every point of \(S\).
The condition \(X_p \in T_pS\) is naturally tested against functions vanishing on \(S\). Tangent
vectors to \(S\) annihilate every function flat along \(S\), and any tangent vector at \(p\) failing
to annihilate some such function points away from \(S\). Globalizing over all \(p \in S\), an
ambient vector field is tangent to \(S\) exactly when the function \(Xf\) vanishes on \(S\) whenever
\(f\) does.
Proposition (Tangency Through Defining Functions)
Let \(M\) be a smooth manifold, \(S \subseteq M\) an embedded submanifold, and
\(X \in \mathfrak{X}(M)\). Then \(X\) is tangent to \(S\) if and only if for every
\(f \in C^\infty(M)\) with \(f|_S = 0\), the function \(Xf\) satisfies \((Xf)|_S = 0\).
Proof:
At each point \(p \in S\), the
tangent space as annihilator
characterizes membership in \(T_pS\) as the vanishing of the action on every smooth function
flat along \(S\). A tangent vector \(v \in T_pM\) lies in \(T_pS\) if and only if \(vf = 0\) for
every \(f \in C^\infty(M)\) with \(f|_S = 0\). Applying this to \(v = X_p\) for each
\(p \in S\),
\[
X_p \in T_pS \quad \text{for every } p \in S
\iff
(Xf)(p) = X_p f = 0 \quad \text{for every } p \in S \text{ and every such } f ,
\]
and the right-hand condition is precisely \((Xf)|_S = 0\) for every \(f \in C^\infty(M)\) with
\(f|_S = 0\), which is the statement of the proposition.
Restricting a tangent vector field
Once a vector field on \(M\) is known to be tangent to \(S\), its values along \(S\) themselves form
tangent vectors to \(S\) at every point. The next proposition records that these values assemble
into a smooth vector field on \(S\), and that the original field is the \(\iota\)-related image of
this restriction under the inclusion \(\iota : S \hookrightarrow M\). This is the submanifold
counterpart of the diffeomorphism pushforward. Existence and smoothness are automatic, with the
inclusion playing the role of the diffeomorphism.
Proposition (Restricting Vector Fields to Submanifolds)
Let \(M\) be a smooth manifold, \(S \subseteq M\) an immersed submanifold, and
\(\iota : S \hookrightarrow M\) the inclusion map. If \(Y \in \mathfrak{X}(M)\) is tangent to
\(S\), there exists a unique smooth vector field \(X \in \mathfrak{X}(S)\), denoted \(Y|_S\),
that is \(\iota\)-related to \(Y\): for every \(p \in S\),
\[
d\iota_p (X_p) = Y_p .
\]
Proof:
Uniqueness and pointwise existence.
For each \(p \in S\), the hypothesis
that \(Y\) is tangent to \(S\) gives \(Y_p \in T_pS\), and \(d\iota_p : T_pS \to T_pM\) is the
inclusion of \(T_pS\) into \(T_pM\), injective because \(\iota\) is a
smooth immersion.
There is therefore a unique \(X_p \in T_pS\) with \(d\iota_p(X_p) = Y_p\), and \(X_p\) is in
fact \(Y_p\) viewed as an element of \(T_pS\). The assignment \(p \mapsto X_p\) is a rough
vector field on \(S\), and it is the unique candidate for a smooth one with the required
property. It remains to verify smoothness.
Smoothness.
Smoothness is a local question on \(S\), so we work near an
arbitrary point \(p \in S\). Since an
immersed submanifold is locally embedded,
there is a neighborhood \(V\) of \(p\) in \(S\) on which the inclusion \(V \hookrightarrow M\)
is an embedding. Choose a
slice chart
\((U, (x^i))\) for \(V\) in \(M\), centered at \(p\), in which \(V \cap U\) is the subset where
\(x^{k+1} = \cdots = x^n = 0\), with \(k = \dim S\). Such a chart exists because \(V\) is
embedded, by the
local slice criterion,
which also gives that the restrictions \((x^1, \dots, x^k)\) of the first \(k\) coordinates to
\(V \cap U\) are local coordinates for \(S\). Write the ambient field in these coordinates as
\[
Y = Y^1\, \frac{\partial}{\partial x^1} + \cdots + Y^n\, \frac{\partial}{\partial x^n}
\quad \text{on } U ,
\]
with smooth component functions \(Y^i \in C^\infty(U)\). The hypothesis that \(Y\) is tangent to
\(S\) means \(Y_q \in T_qS\) for every \(q \in V \cap U\), and in the slice chart \(T_qS\) is
the subspace spanned by \(\partial/\partial x^1|_q, \dots, \partial/\partial x^k|_q\). The
components \(Y^{k+1}, \dots, Y^n\) therefore vanish on \(V \cap U\).
It follows from the pointwise construction in the first part of the proof that \(X\) has the
coordinate representation
\[
X = Y^1\bigl|_{V \cap U}\, \frac{\partial}{\partial x^1}
+ \cdots
+ Y^k\bigl|_{V \cap U}\, \frac{\partial}{\partial x^k}
\quad \text{on } V \cap U ,
\]
in the coordinates \((x^1, \dots, x^k)\) for \(S\). Each coefficient is the restriction of a
smooth function on \(U\) and is therefore smooth on \(V \cap U\), so \(X\) is smooth on
\(V \cap U\). Since every point of \(S\) has such a coordinate neighborhood, \(X\) is smooth on
\(S\).
When the submanifold \(S\) has boundary, the construction goes through with a slice chart in which
\(V \cap U\) is cut out by \(x^{k+1} = \cdots = x^n = 0\) together with \(x^k \geq 0\), a
half-slice rather than an ordinary slice. The convention adopted here keeps boundaries in view only where
they alter the argument, and the slice-chart construction above is the form we use by default.
The Lie Bracket
We now turn to the second operation announced at the start of this page, combining two smooth vector
fields to produce a third. The natural first attempt is to compose the two as differential operators
on \(C^\infty(M)\), in the
derivation picture.
That attempt fails, because the composition \(XY\) of two vector fields, viewed as operators, is not
in general a derivation, hence not a vector field. The antisymmetrization \(XY - YX\) does succeed,
and the resulting operation is the Lie bracket. It equips \(\mathfrak{X}(M)\) with the structure of
a Lie algebra, made explicit in the next section.
Why composition alone fails
A smooth vector field \(X\) acts on smooth functions, sending \(f \in C^\infty(M)\) to
\(Xf \in C^\infty(M)\). Composing two such actions yields a linear operator \(f \mapsto X(Yf)\),
which we write \(XYf\). This operator is \(\mathbb{R}\)-linear, but it is not a derivation. The
product rule fails, with second-order cross terms appearing that have no first-order interpretation
as the directional derivative of a single vector field.
Example (Failure of the Product Rule for \(XY\)):
On \(\mathbb{R}^2\), let \(X = \partial/\partial x\) and \(Y = x\, \partial/\partial y\), and let
\(f(x, y) = x\), \(g(x, y) = y\). Computing the left-hand side of the would-be product rule,
\[
\begin{align*}
Y(fg) &= x \cdot \frac{\partial(xy)}{\partial y} = x \cdot x = x^2 , \\\\
XY(fg) &= \frac{\partial(x^2)}{\partial x} = 2x.
\end{align*}
\]
For the right-hand side,
\[
\begin{align*}
Yg &= x \cdot \frac{\partial y}{\partial y} = x ,
\quad XYg = \frac{\partial x}{\partial x} = 1 , \\\\
Yf &= x \cdot \frac{\partial x}{\partial y} = 0 ,
\quad XYf = 0 ,
\end{align*}
\]
and so
\[
f \cdot XYg + g \cdot XYf = x \cdot 1 + y \cdot 0 = x.
\]
Since \(2x \neq x\), the operator \(XY\) does not satisfy the product rule, and is not a derivation of
\(C^\infty(\mathbb{R}^2)\).
The asymmetry between the two sides of the failed identity has a clean structure. The missing terms
are precisely the ones that would appear from applying the operators in the reverse order. The
cancellation that gives a derivation is therefore the difference \(XY - YX\).
The Lie bracket
Definition: Lie Bracket of Vector Fields
Let \(M\) be a smooth manifold with or without boundary, and let \(X, Y \in \mathfrak{X}(M)\) be
smooth vector fields. The Lie bracket of \(X\) and \(Y\) is the linear operator
\([X, Y] : C^\infty(M) \to C^\infty(M)\) defined by
\[
[X, Y] f = X (Yf) - Y (Xf) \quad \text{for } f \in C^\infty(M) .
\]
The notation matches the
matrix Lie bracket
\([A, B] = AB - BA\) used earlier in the linear-algebra track, and the match is more than cosmetic.
When the manifold is a matrix Lie group, the bracket of vector fields adapted to its
left-translation structure will reduce exactly to the matrix commutator at the identity. That
correspondence is developed in a later page of this series. For the moment we treat \([X, Y]\)
purely as an operator on smooth functions and show that it is in fact a vector field.
The bracket is a smooth vector field
Lemma (The Lie Bracket Is a Smooth Vector Field)
For any \(X, Y \in \mathfrak{X}(M)\), the operator \([X, Y]\) is a derivation of \(C^\infty(M)\),
and is therefore the action on functions of a uniquely determined smooth vector field on \(M\),
which we identify with \([X, Y]\).
Proof:
Linearity of \([X, Y]\) over \(\mathbb{R}\) is inherited from the linearity of \(X\) and \(Y\)
and the linearity of composition. It remains to verify the product rule for \([X, Y]\). For any
\(f, g \in C^\infty(M)\), applying the product rule for \(X\) and \(Y\) twice gives
\[
\begin{align*}
X\bigl( Y(fg) \bigr)
&= X\bigl( f\, Yg + g\, Yf \bigr) \\\\
&= Xf \cdot Yg + f \cdot XYg + Xg \cdot Yf + g \cdot XYf ,
\end{align*}
\]
and symmetrically
\[
\begin{align*}
Y\bigl( X(fg) \bigr)
&= Y\bigl( f\, Xg + g\, Xf \bigr) \\\\
&= Yf \cdot Xg + f \cdot YXg + Yg \cdot Xf + g \cdot YXf .
\end{align*}
\]
Subtracting, the cross terms cancel in pairs. The product of real-valued smooth functions is
commutative, so \(Xf \cdot Yg = Yg \cdot Xf\) and \(Xg \cdot Yf = Yf \cdot Xg\), leaving
\[
\begin{align*}
[X, Y](fg)
&= X(Y(fg)) - Y(X(fg)) \\\\
&= f \cdot (XYg - YXg) + g \cdot (XYf - YXf) \\\\
&= f \cdot [X, Y] g + g \cdot [X, Y] f .
\end{align*}
\]
Hence \([X, Y]\) is a derivation of \(C^\infty(M)\). By the
identification of smooth vector fields with derivations,
there is a unique smooth vector field on \(M\) whose action on functions is \([X, Y]\), and we
denote that vector field by \([X, Y]\) as well.
Pointwise formula and computational practicality
Evaluating the bracket as a vector field at a point yields
\[
[X, Y]_p f = X_p (Yf) - Y_p (Xf) ,
\]
a tangent vector at \(p\) acting on functions defined near \(p\). This pointwise formula is the most
direct definition, but it is awkward in computations. Applying it to a specific \(f\) produces
second derivatives of \(f\), which must then be shown to cancel by hand. A coordinate formula in
which the cancellation has already been carried out gives a more tractable expression.
The coordinate formula
Proof:
Both sides are smooth vector fields on \(U\), and equality of vector fields is a local question
on which both sides act on any \(f \in C^\infty(U)\). It therefore suffices to apply both sides
to a smooth function \(f\) and show that the resulting real-valued functions agree on \(U\).
Expanding by the product rule,
\[
\begin{align*}
[X, Y] f
&= X(Yf) - Y(Xf) \\\\
&= X^i\, \frac{\partial}{\partial x^i}
\left( Y^j\, \frac{\partial f}{\partial x^j} \right)
- Y^j\, \frac{\partial}{\partial x^j}
\left( X^i\, \frac{\partial f}{\partial x^i} \right) \\\\
&= X^i\, \frac{\partial Y^j}{\partial x^i}\, \frac{\partial f}{\partial x^j}
+ X^i Y^j\, \frac{\partial^2 f}{\partial x^i\, \partial x^j} \\\\
&\quad - Y^j\, \frac{\partial X^i}{\partial x^j}\, \frac{\partial f}{\partial x^i}
- Y^j X^i\, \frac{\partial^2 f}{\partial x^j\, \partial x^i} .
\end{align*}
\]
The two second-derivative terms are equal. We have \(X^i Y^j = Y^j X^i\) by commutativity of
multiplication of real-valued functions, and
\(\partial^2 f / \partial x^i\, \partial x^j = \partial^2 f / \partial x^j\, \partial x^i\) for
any \(f \in C^\infty(U)\). They cancel.
Renaming the dummy index \(i \leftrightarrow j\) in the third surviving term turns it into
\(- Y^i\, (\partial X^j / \partial x^i)\, \partial f/\partial x^j\), bringing both surviving
terms onto the common factor \(\partial f/\partial x^j\). The result is
\[
[X, Y] f
= \left( X^i\, \frac{\partial Y^j}{\partial x^i}
- Y^i\, \frac{\partial X^j}{\partial x^i} \right)
\frac{\partial f}{\partial x^j} ,
\]
valid for every \(f \in C^\infty(U)\), which is the operator equality of the proposition.
Recognizing \(X^i\, \partial Y^j/\partial x^i = X Y^j\) and similarly for \(Y X^j\) gives the
concise form.
Brackets of coordinate vector fields
A trivial special case of the coordinate formula is the bracket of two coordinate vector fields,
whose component functions are constants.
Proposition (Coordinate Vector Fields Commute)
In any smooth chart \((U, (x^i))\), the coordinate vector fields satisfy
\[
\left[ \frac{\partial}{\partial x^i},\ \frac{\partial}{\partial x^j} \right] = 0
\quad \text{for all } i, j .
\]
Proof:
The \(k\)th component function of \(\partial/\partial x^i\) in the chart \((U, (x^i))\) is the
Kronecker delta \(\delta^k_i\), a constant, so its partial derivatives in any of the coordinate
directions vanish. Substituting into the coordinate formula above, every coefficient is zero, so
the bracket vanishes.
The fact that coordinate vector fields commute is a restatement, at the level of the bracket, of the
commutativity of mixed partial derivatives of smooth functions. A geometric interpretation of the
Lie bracket as the obstruction to commuting flows of two vector fields is available in a later stage
of this development. For now, the formula is the computational tool we need.
A worked example
Example (Lie Bracket of Two Vector Fields on \(\mathbb{R}^3\)):
On \(\mathbb{R}^3\), let
\[
X = x\, \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + x(y + 1)\, \frac{\partial}{\partial z} ,
\quad
Y = \frac{\partial}{\partial x} + y\, \frac{\partial}{\partial z} ,
\]
with component functions \(X^x = x\), \(X^y = 1\), \(X^z = x(y+1)\) and \(Y^x = 1\), \(Y^y = 0\),
\(Y^z = y\). Applying the concise coordinate formula to each component:
\[
\begin{align*}
X Y^x - Y X^x
&= X(1) - Y(x)
= 0 - 1
= -1 ,\\\\
X Y^y - Y X^y
&= X(0) - Y(1)
= 0 - 0
= 0 ,\\\\
X Y^z - Y X^z
&= X(y) - Y\bigl( x(y + 1) \bigr) .
\end{align*}
\]
For the last entry, \(X(y) = X^y \cdot 1 = 1\) since \(\partial y/\partial y = 1\), and
\(Y(x(y+1)) = \partial(x(y+1))/\partial x + y\, \partial(x(y+1))/\partial z = (y + 1) + 0 = y + 1\),
so \(X Y^z - Y X^z = 1 - (y + 1) = -y\). Assembling,
\[
[X, Y]
= -\frac{\partial}{\partial x} - y\, \frac{\partial}{\partial z} .
\]
Properties of the Bracket
The Lie bracket of vector fields is bilinear, antisymmetric, and satisfies a Jacobi identity, making
\((\mathfrak{X}(M), [\,\cdot\,, \cdot\,])\) into a Lie algebra over \(\mathbb{R}\). It also
satisfies a product rule with respect to the action of smooth functions on each argument. Beyond
these identities purely internal to a fixed manifold, the bracket interacts cleanly with smooth maps
and with restriction to submanifolds. It commutes with the operations of \(F\)-relatedness,
pushforward by a diffeomorphism, and restriction along an inclusion.
Internal identities of the bracket
Proposition (Properties of the Lie Bracket)
Let \(M\) be a smooth manifold with or without boundary. For all
\(X, Y, Z \in \mathfrak{X}(M)\), \(a, b \in \mathbb{R}\), and \(f, g \in C^\infty(M)\), the
Lie bracket satisfies the following identities.
(a) Bilinearity.
\[
[aX + bY, Z] = a[X, Z] + b[Y, Z] ,
\quad
[Z, aX + bY] = a[Z, X] + b[Z, Y] .
\]
(b) Antisymmetry.
\[
[X, Y] = -[Y, X] .
\]
(c) Jacobi identity.
\[
\bigl[ X, [Y, Z] \bigr] + \bigl[ Y, [Z, X] \bigr] + \bigl[ Z, [X, Y] \bigr] = 0 .
\]
(d) Product rule with respect to \(C^\infty(M)\)-scaling.
\[
[fX, gY] = fg\, [X, Y] + (f\, Xg)\, Y - (g\, Yf)\, X .
\]
Proof:
(a) and (b) follow immediately from the operator definition
\([X, Y] f = X(Yf) - Y(Xf)\). The \(\mathbb{R}\)-linearity of each entry is the linearity of
\(X\) and \(Y\) acting on functions, and \([X, Y] = X Y - Y X = -(YX - XY) = -[Y, X]\) is the
antisymmetry of the commutator.
(c) Jacobi identity. Applying the left-hand side to an arbitrary
\(f \in C^\infty(M)\) and expanding each outer bracket gives
\[
\begin{align*}
\bigl[ X, [Y, Z] \bigr] f
&= X\bigl( [Y, Z] f \bigr) - [Y, Z] (Xf) \\\\
&= X\bigl( Y(Zf) - Z(Yf) \bigr) - Y\bigl( Z(Xf) \bigr) + Z\bigl( Y(Xf) \bigr) \\\\
&= XYZf - XZYf - YZXf + ZYXf ,
\end{align*}
\]
and cyclically
\[
\begin{align*}
\bigl[ Y, [Z, X] \bigr] f
&= YZXf - YXZf - ZXYf + XZYf ,\\\\
\bigl[ Z, [X, Y] \bigr] f
&= ZXYf - ZYXf - XYZf + YXZf .
\end{align*}
\]
Adding the three lines, each of the six monomials \(XYZf\), \(XZYf\), \(YXZf\), \(YZXf\),
\(ZXYf\), \(ZYXf\) appears once with a positive sign and once with a negative sign and so
cancels. The total is zero for every \(f\), proving the identity.
(d) Product rule. Applying both sides to an arbitrary \(h \in C^\infty(M)\),
the left-hand side expands by the product rule for \(X\) and \(Y\) acting on functions as
\[
\begin{align*}
[fX, gY] h
&= fX(gYh) - gY(fXh) \\\\
&= f \cdot \bigl( Xg \cdot Yh + g \cdot XYh \bigr)
- g \cdot \bigl( Yf \cdot Xh + f \cdot YXh \bigr) \\\\
&= fg \cdot \bigl( XYh - YXh \bigr)
+ (fXg) \cdot Yh
- (gYf) \cdot Xh \\\\
&= fg \cdot [X, Y] h + (fXg)\, Yh - (gYf)\, Xh ,
\end{align*}
\]
which is the right-hand side applied to \(h\). Since the equality holds for every \(h\), the
two operators agree as vector fields.
Properties (a), (b), and (c) together say that the pair \((\mathfrak{X}(M), [\,\cdot\,, \cdot\,])\)
is a Lie algebra over the field of real numbers, that is, an \(\mathbb{R}\)-vector space equipped
with an antisymmetric bilinear operation satisfying the Jacobi identity. Property (d) records how
the bracket interacts with the \(C^\infty(M)\)-module structure on \(\mathfrak{X}(M)\) established
earlier. It is the price one pays for using functional rather than scalar coefficients, and it
controls many later computations where smooth function factors appear inside brackets.
Naturality under smooth maps
Beyond these internal identities, the bracket interacts cleanly with smooth maps. If both inputs of
a bracket are \(F\)-related across a smooth map, then so is the bracket itself. This compatibility
is what allows Lie-algebra constructions on different manifolds to be transported between them, and
it is the key technical fact used later when matrix groups and abstract Lie groups are reconciled.
Proposition (Naturality of the Lie Bracket)
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary, and let
\(X_1, X_2 \in \mathfrak{X}(M)\) and \(Y_1, Y_2 \in \mathfrak{X}(N)\). If \(X_i\) is
\(F\)-related to \(Y_i\) for \(i = 1, 2\), then \([X_1, X_2]\) is \(F\)-related to
\([Y_1, Y_2]\).
Proof:
By the
function-level characterization of \(F\)-relatedness,
\(X_i\) being \(F\)-related to \(Y_i\) means
\[
X_i (f \circ F) = (Y_i f) \circ F
\quad \text{for every open } V \subseteq N \text{ and every } f \in C^\infty(V) .
\]
Applying this with \(i = 2\) and then with \(i = 1\) to the function \(Y_2 f\) gives
\[
X_1 X_2 (f \circ F)
= X_1 \bigl( (Y_2 f) \circ F \bigr)
= (Y_1 Y_2 f) \circ F,
\]
and symmetrically \(X_2 X_1 (f \circ F) = (Y_2 Y_1 f) \circ F\). Subtracting,
\[
\begin{align*}
[X_1, X_2] (f \circ F)
&= X_1 X_2 (f \circ F) - X_2 X_1 (f \circ F) \\\\
&= (Y_1 Y_2 f) \circ F - (Y_2 Y_1 f) \circ F \\\\
&= \bigl( [Y_1, Y_2] f \bigr) \circ F.
\end{align*}
\]
Since this holds for every \(f\), the function-level characterization applied in the reverse
direction yields that \([X_1, X_2]\) is \(F\)-related to \([Y_1, Y_2]\).
The term naturality reflects a categorical perspective relating pairs
\((M, \mathfrak{X}(M))\) and maps between them. We use the name without pursuing the categorical
formalism here.
Bracket and pushforward
When the smooth map is a diffeomorphism, every vector field on the source has a uniquely determined
\(F\)-related field on the target, namely the pushforward. Naturality then becomes a statement that
the pushforward commutes with the Lie bracket.
Corollary (Pushforward Commutes with the Bracket)
Let \(F : M \to N\) be a diffeomorphism and \(X_1, X_2 \in \mathfrak{X}(M)\). Then
\[
F_* [X_1, X_2] = [F_* X_1, F_* X_2].
\]
Proof:
By the
definition of the pushforward,
\(F_* X_i\) is the unique smooth vector field on \(N\) that is \(F\)-related to \(X_i\). By
naturality of the bracket, \([X_1, X_2]\) is \(F\)-related to \([F_* X_1, F_* X_2]\). Since
\(F\) is a diffeomorphism, the \(F\)-related vector field on \(N\) for the given input
\([X_1, X_2]\) is unique and is denoted \(F_* [X_1, X_2]\). Therefore the two are equal.
Brackets along a submanifold
The submanifold counterpart of pushforward is the restriction of a tangent field along an inclusion.
Naturality applied to the inclusion map shows that the bracket of two ambient vector fields tangent
to a submanifold is itself tangent to that submanifold.
Corollary (Brackets of Tangent Vector Fields Are Tangent)
Let \(M\) be a smooth manifold, \(S \subseteq M\) an
immersed submanifold with or without boundary,
and \(Y_1, Y_2 \in \mathfrak{X}(M)\) both tangent to \(S\). Then the Lie bracket
\([Y_1, Y_2]\) is also tangent to \(S\).
Proof:
Let \(\iota : S \hookrightarrow M\) be the inclusion. By the
restriction proposition
for vector fields tangent to a submanifold, there exist smooth vector fields
\(X_1, X_2 \in \mathfrak{X}(S)\) with \(X_i\) \(\iota\)-related to \(Y_i\). By naturality of the
bracket, \([X_1, X_2] \in \mathfrak{X}(S)\) is \(\iota\)-related to \([Y_1, Y_2]\). This means
precisely that for every \(p \in S\),
\[
d\iota_p\bigl( [X_1, X_2]_p \bigr) = [Y_1, Y_2]_p,
\]
so \([Y_1, Y_2]_p\) lies in the image of \(d\iota_p\), which is \(T_pS\) regarded as a subspace
of \(T_pM\). Hence \([Y_1, Y_2]\) is tangent to \(S\) at every point of \(S\).
Controllability and Symbolic Brackets
The two threads we close this page on illustrate where the bracket operates outside pure
differential geometry. The first is the role of the Lie bracket in control theory, where it measures
the reachable configurations of mechanical systems governed by a fixed collection of allowable
velocities. The second is the practical one. The coordinate formula for the bracket is exactly the
rule by which symbolic algebra systems compute brackets of vector fields, making the derivation
picture established on the previous page directly available to computation. The further application,
the connection between the bracket on \(\mathfrak{X}(G)\) of a Lie group and the matrix commutator,
we defer to a separate page where the requisite group structure is taken up.
The Lie bracket and controllability
A mechanical system whose state is a configuration \(q\) on a smooth manifold \(Q\) is often
described by a driftless input-affine equation
\[
\dot{q}(t) = \sum_{i=1}^m u_i(t)\, f_i\bigl( q(t) \bigr) ,
\]
where \(f_1, \dots, f_m \in \mathfrak{X}(Q)\) are smooth vector fields on the configuration manifold
and \(u_1, \dots, u_m\) are real-valued control inputs the operator chooses as functions of time.
Each \(f_i\) is the instantaneous velocity contributed by activating the \(i\)th control. Examples
include a wheeled robot whose two controls are the rolling speed and steering rate, a satellite
whose controls are the momentum-wheel torques, and a robotic arm whose controls are the
joint-angular velocities.
At each configuration \(q\), the available instantaneous velocities span a linear subspace
\[
\Delta_q = \mathrm{span}\{ f_1(q), \dots, f_m(q) \} \subseteq T_q Q.
\]
When \(\Delta_q\) is a proper subspace of \(T_q Q\), the system cannot move in directions outside
\(\Delta_q\) at any given instant. The question is whether such directions are nonetheless reachable
by composing sequences of small control moves. The answer is governed by the Lie bracket. If at some
configuration \(q\) the bracket \([f_i, f_j](q)\) lies outside \(\Delta_q\), then by alternating
short pulses of \(f_i\) and \(f_j\) the system can produce a net motion in the bracket direction,
even though that direction was unavailable instantaneously.
Iterating this construction produces a nested family of subspaces of \(T_q Q\). The iteration allows
brackets of brackets, brackets of those, and so on. If the resulting iterated brackets eventually
span all of \(T_q Q\) at every \(q\), the system is said to satisfy the bracket-generating
condition, and a classical theorem of control theory then guarantees that any configuration is
reachable from any other along a sequence of control trajectories. Spelling out this theorem, and
proving it, requires the language of flows on manifolds developed in the next stage of this theory.
The relevance of the bracket to the question, however, is already visible here. The bracket measures
the failure of two flows to commute, and the failure to commute is what allows compound motions to
generate directions inaccessible to either flow alone.
Coordinate formula and symbolic computation
Beyond its theoretical role, the coordinate formula proved earlier on this page is the rule by
which symbolic algebra systems for differential geometry compute brackets. Given vector fields
\(X = X^i \partial/\partial x^i\) and \(Y = Y^j \partial/\partial x^j\) presented by their
component functions in a chart, the bracket
\[
[X, Y]
= \left( X^i\, \frac{\partial Y^j}{\partial x^i}
- Y^i\, \frac{\partial X^j}{\partial x^i} \right)
\frac{\partial}{\partial x^j}
\]
reduces the operation to a finite collection of symbolic differentiations and arithmetic operations
on the component functions. Modern differential-geometry packages in computer algebra systems
implement this formula as the basic bracket operation on vector fields, and the worked example on
\(\mathbb{R}^3\) given earlier on this page was, in effect, an instance of the same computation a
symbolic system would carry out automatically.
The general point is that the operator description of vector fields developed on the previous page,
together with the bracket formula derived on this one, fits naturally with the standard data
structures of computer algebra. A vector field is a tuple of smooth component functions, the action
on a function is a coordinate-by-coordinate differentiation, and the bracket is one further
differentiation step. The same description supports the algorithms used in geometric integrators, in
symbolic Hamiltonian mechanics, and in computer-aided derivations within control and robotics.
The next page in this development specializes the bracket to the setting where the manifold itself
carries a Lie group structure, and singles out the vector fields invariant under the group's
left-translation action. That class of vector fields is closed under the bracket,
finite-dimensional, and entirely determined by its values at the identity. These properties together
turn it into a finite-dimensional Lie algebra, and they recover the matrix commutator on
\(\mathfrak{gl}(n, \mathbb{R})\) as a special case of the construction on this page.