The Need for Local Flows
The constructions of the previous development tied together two pictures of a smooth
vector field. On the geometric side, each smooth vector field admits an integral curve
through every point, defined on some open interval around \(t = 0\). On the dynamical
side, a smooth global flow on \(M\) is determined entirely by its
infinitesimal generator,
a smooth vector field whose integral curves are the trajectories of the flow. The
natural question is whether the correspondence runs in the opposite direction as well:
given a smooth vector field, can we always produce a smooth global flow whose
infinitesimal generator it is?
The answer is no, and the obstruction is not a defect of the abstract machinery but a
feature of the underlying differential equations. There exist smooth vector fields
whose integral curves cannot be continued for all real time — either because the
trajectory runs out of the manifold in finite time, or because a coordinate component
of the trajectory becomes unbounded. Two examples on subsets of \(\mathbb{R}^2\) make
both failure modes explicit.
Two Counterexamples
Example (A Hole in the Manifold):
Let \(M = \mathbb{R}^2 \setminus \{0\}\) with the standard coordinates inherited
from the plane, and let \(V = \partial / \partial x\). The integral curve of \(V\)
starting at the point \((-1, 0)\) is given by \(\gamma(t) = (t - 1, \, 0)\) for
\(t \in (-\infty, 1)\); at \(t = 1\) the curve would reach the origin, which has
been deleted from \(M\). Suppose, for contradiction, that
\(\tilde\gamma : (-\infty, 1 + \varepsilon) \to M\) is a continuous extension of
\(\gamma\) past \(t = 1\); by definition of extension,
\(\tilde\gamma(t) = \gamma(t) = (t - 1, 0)\) for \(t < 1\). Composing with the
inclusion \(M \hookrightarrow \mathbb{R}^2\) and taking the limit \(t \nearrow 1\)
in \(\mathbb{R}^2\) gives \(\tilde\gamma(t) \to (0, 0)\). Limits in
\(\mathbb{R}^2\) are unique and \(\tilde\gamma\) is continuous, so
\(\tilde\gamma(1) = (0, 0) \notin M\), contradicting the codomain of
\(\tilde\gamma\). The integral curve through \((-1, 0)\) is therefore maximally
defined on \((-\infty, 1)\), and \(V\) does not generate a global flow on \(M\).
Example (Finite-Time Blow-Up):
Let \(M = \mathbb{R}^2\) and \(W = x^2 \, \partial / \partial x\). The integral
curve starting at \((1, 0)\) is the solution of \(\dot x = x^2\) with \(x(0) = 1\),
which by separation of variables is \(x(t) = 1 / (1 - t)\). Thus
\(\gamma(t) = \bigl( 1 / (1 - t), \, 0 \bigr)\) is defined on \((-\infty, 1)\) and
cannot be continued to \(t = 1\): its \(x\)-coordinate diverges as \(t \nearrow 1\).
Here the manifold is complete (it is the whole plane), and the failure is not a
geometric defect but a consequence of the right-hand side of the differential
equation growing too rapidly along the trajectory.
Together, the two examples display a general phenomenon: a smooth vector field whose
integral curves are perfectly well-defined on each maximal existence interval need not
extend to a global flow on \(\mathbb{R} \times M\). The right framework for accommodating
this is to allow the time variable to range over an open subset of
\(\mathbb{R} \times M\) that depends on the starting point.
Flow Domains and Local Flows
The structure that emerges from the two examples is captured by the following
definition.
Definition: Flow Domain
A flow domain on a smooth manifold \(M\) is an open subset
\(\mathcal{D} \subseteq \mathbb{R} \times M\) with the property that, for every
\(p \in M\), the set
\[
\mathcal{D}^{(p)} = \{ t \in \mathbb{R} : (t, p) \in \mathcal{D} \}
\]
is an open interval containing \(0\).
Definition: Flow on a Flow Domain
A flow (also called a local flow or local
one-parameter group action) on \(M\) is a continuous map
\(\theta : \mathcal{D} \to M\), where \(\mathcal{D}\) is a flow domain on \(M\),
satisfying
\[
\theta(0, p) = p \qquad \text{for all } p \in M ,
\]
and the group law
\[
\theta\bigl( t, \theta(s, p) \bigr) = \theta(t + s, p)
\]
whenever both sides are defined — that is, whenever \(s \in \mathcal{D}^{(p)}\) and
\(t \in \mathcal{D}^{(\theta(s, p))}\) and the sum \(t + s\) likewise lies in
\(\mathcal{D}^{(p)}\).
For each \(t \in \mathbb{R}\), the partial map \(\theta_t\) is defined on the open set
\[
M_t = \{ p \in M : (t, p) \in \mathcal{D} \} ,
\]
where it acts as \(\theta_t(p) = \theta(t, p)\). The group law has a clean
interpretation in terms of these maps: where defined, \(\theta_t \circ \theta_s = \theta_{t + s}\),
and \(\theta_0 = \mathrm{Id}_M\) globally. A smooth flow on \(\mathcal{D}\) is one for
which the underlying map \(\theta\) is smooth as a map of manifolds.
For each \(p \in M\), the trajectory of \(p\) under the flow is the smooth curve
\(\theta^{(p)} : \mathcal{D}^{(p)} \to M\) defined by \(\theta^{(p)}(t) = \theta(t, p)\),
and the infinitesimal generator of a smooth flow is defined as before by
\(V_p = (\theta^{(p)})'(0) \in T_p M\). When \(\mathcal{D} = \mathbb{R} \times M\) and
each \(\mathcal{D}^{(p)} = \mathbb{R}\), the flow is global in the sense of the
previous development; the cases of the two examples above are decidedly not of this
form.
Maximal Integral Curves and Maximal Flows
Among the integral curves through a fixed point, one is naturally distinguished: the
one whose domain cannot be enlarged.
Definition: Maximal Integral Curve
Let \(V\) be a smooth vector field on \(M\). A
integral curve
\(\gamma : J \to M\) of \(V\) is maximal if it admits no
extension to an integral curve of \(V\) on an open interval strictly containing
\(J\).
Definition: Maximal Flow
A flow \(\theta : \mathcal{D} \to M\) is maximal if it admits no
extension to a flow on a flow domain strictly larger than \(\mathcal{D}\).
These definitions describe the right way to package the integral curves of a smooth
vector field: each starting point should be assigned its uniquely largest existence
interval, and the resulting curves should be collected into a single map on the largest
flow domain compatible with the dynamics. The central result of this development, the
fundamental theorem on flows, asserts that this packaging always succeeds.
The Infinitesimal Generator of a Smooth Flow
The first step toward that result is a small extension of an earlier observation. The
proposition that a smooth global flow has a smooth infinitesimal generator carries over
verbatim to the local setting: the proof used only the openness of the time domain,
never the assumption that the flow was defined for all real \(t\).
Proposition (Infinitesimal Generator of a Smooth Flow)
Let \(\theta : \mathcal{D} \to M\) be a smooth flow on a smooth manifold \(M\), and
let \(V\) be its infinitesimal generator. Then \(V\) is a smooth vector field on
\(M\), and for each \(p \in M\) the trajectory \(\theta^{(p)}\) is an integral
curve of \(V\) on \(\mathcal{D}^{(p)}\).
The argument is essentially the same as for a smooth global flow. For smoothness, one
checks that for every open \(U \subseteq M\) and \(f \in C^\infty(U)\), the function
\(Vf\) on \(U\) equals the partial derivative of \(f \circ \theta\) with respect to
\(t\) at \((0, p)\), which is smooth in \(p\) because \(\mathcal{D}\) is open and
\(\theta\) is smooth; the
smoothness criterion for vector fields via their action on functions
then identifies \(V\) as smooth. For the integral-curve property, the group law gives
\(\theta^{(p)}(t + t_0) = \theta^{(\theta(t_0, p))}(t)\) on an open neighborhood of
\(t = 0\) in \(\mathcal{D}^{(p)} - t_0\); differentiating at \(t = 0\) yields
\((\theta^{(p)})'(t_0) = V_{\theta(t_0, p)}\), which is the integral-curve condition
at \(t_0\).
With these definitions in hand, the central question becomes: does every smooth vector
field on \(M\) arise as the infinitesimal generator of a unique maximal smooth flow on
some flow domain \(\mathcal{D}\)? The next section answers this question
affirmatively.
The Fundamental Theorem on Flows
The fundamental theorem of this section asserts that every smooth vector field on a
smooth manifold determines a unique maximal smooth flow, and identifies the structural
properties of that flow. The statement has three parts: maximality and uniqueness of
the integral curve through each point, a group law tying the maximal existence
intervals of different starting points, and a diffeomorphism property of the time-\(t\)
maps. The proof is correspondingly organized in stages, the first three of which
construct the maximal flow as a set-theoretic object, the fourth of which proves it is
smooth on an open domain, and the fifth of which deduces the diffeomorphism property
as an immediate corollary.
Theorem (Fundamental Theorem on Flows)
Let \(V\) be a smooth vector field on a smooth manifold \(M\). There is a unique
smooth maximal flow \(\theta : \mathcal{D} \to M\) whose infinitesimal generator
is \(V\). This flow has the following properties:
(a) For each \(p \in M\), the trajectory
\(\theta^{(p)} : \mathcal{D}^{(p)} \to M\) is the unique maximal integral curve of
\(V\) starting at \(p\).
(b) If \(s \in \mathcal{D}^{(p)}\), then
\(\mathcal{D}^{(\theta(s, p))} = \mathcal{D}^{(p)} - s\), the set obtained by
subtracting \(s\) from every element of \(\mathcal{D}^{(p)}\).
(c) For each \(t \in \mathbb{R}\), the set \(M_t\) is open in
\(M\), and the map \(\theta_t : M_t \to M_{-t}\) is a diffeomorphism with inverse
\(\theta_{-t}\).
The proof divides naturally into five steps. We carry out each in turn.
Step 1: Uniqueness of Integral Curves
The first step rules out a phenomenon the local existence statement leaves open: two
distinct integral curves passing through the same point with the same velocity. Once
this is excluded, the maximal integral curve through a point is well-defined.
Lemma (Uniqueness of Integral Curves):
Let \(V\) be a smooth vector field on \(M\), and suppose
\(\gamma, \tilde\gamma : J \to M\) are two integral curves of \(V\) defined on the
same open interval \(J \subseteq \mathbb{R}\). If \(\gamma(t_0) = \tilde\gamma(t_0)\)
for some \(t_0 \in J\), then \(\gamma = \tilde\gamma\) on all of \(J\).
Define
\[
S = \{ t \in J : \gamma(t) = \tilde\gamma(t) \} .
\]
The set \(S\) is nonempty because \(t_0 \in S\) by hypothesis, and \(S\) is closed
in \(J\) because \(\gamma\) and \(\tilde\gamma\) are continuous and \(M\) is
Hausdorff.
We show \(S\) is also open in \(J\). Let \(t_1 \in S\), and set
\(p = \gamma(t_1) = \tilde\gamma(t_1)\). Choose a smooth chart \((U, (x^i))\)
centered at \(p\). On the open subset of \(J\) where both curves lie in \(U\), the
coordinate representations \(\gamma^i\) and \(\tilde\gamma^i\) satisfy the same
autonomous system of ordinary differential equations \(\dot u^i(t) = V^i(u(t))\)
with the same initial condition \(u(t_1) = 0\). The uniqueness statement for
solutions of smooth autonomous systems of ordinary differential equations, part
of the same classical existence-uniqueness-smoothness package invoked earlier for
existence, forces \(\gamma^i \equiv \tilde\gamma^i\) on an open
interval around \(t_1\); equivalently, \(\gamma = \tilde\gamma\) on an open
neighborhood of \(t_1\) in \(J\). Hence \(t_1\) is an interior point of \(S\).
The set \(S\) is therefore a nonempty, open, and closed subset of the connected
space \(J\), so \(S = J\), which is the claim.
Step 2: Construction of the Maximal Integral Curve
With uniqueness in hand, the integral curves through a fixed point can be assembled
into a single maximal one. For each \(p \in M\), let
\[
\mathcal{D}^{(p)} = \bigcup_{\gamma} \mathrm{dom}(\gamma) ,
\]
where the union ranges over all integral curves \(\gamma\) of \(V\) starting at \(p\)
and defined on an open interval containing \(0\). The local existence theorem of the
previous development guarantees that this collection is nonempty, so
\(\mathcal{D}^{(p)}\) is a nonempty union of open intervals all containing \(0\), hence
is itself an open interval containing \(0\).
Define \(\theta^{(p)} : \mathcal{D}^{(p)} \to M\) by setting
\(\theta^{(p)}(t) = \gamma(t)\) for any integral curve \(\gamma\) of \(V\) starting at
\(p\) and defined at \(t\). The Step 1 uniqueness lemma applied to any two such
curves \(\gamma_1, \gamma_2\) shows that they agree on their common domain — both
pass through \(p\) at time \(0\) — so the value \(\theta^{(p)}(t)\) does not depend on
the choice of \(\gamma\). The resulting map \(\theta^{(p)}\) is visibly an integral
curve of \(V\) starting at \(p\): for each \(t \in \mathcal{D}^{(p)}\) and each
\(\gamma\) defined at \(t\), the velocity \((\theta^{(p)})'(t) = \gamma'(t) = V_{\gamma(t)} = V_{\theta^{(p)}(t)}\).
The interval \(\mathcal{D}^{(p)}\) is maximal among the domains of integral curves
starting at \(p\): any such curve has domain contained in \(\mathcal{D}^{(p)}\) by
construction, and any extension of \(\theta^{(p)}\) to a strictly larger open interval
would be an integral curve starting at \(p\), hence its domain would already be
contained in \(\mathcal{D}^{(p)}\), a contradiction. The trajectory \(\theta^{(p)}\)
is therefore the unique maximal integral curve of \(V\) starting at \(p\), and
property (a) of the theorem statement is established.
Step 3: The Group Law
Property (b) records how the maximal existence intervals of different starting points
relate to one another along a single trajectory. The translation lemma of the
integral-curve development is the essential tool.
Fix \(p \in M\) and \(s \in \mathcal{D}^{(p)}\), and set \(q = \theta^{(p)}(s)\). The
goal is the equality \(\mathcal{D}^{(q)} = \mathcal{D}^{(p)} - s\). We prove the two
inclusions in turn.
For one inclusion, consider the curve
\[
\gamma : \mathcal{D}^{(p)} - s \to M , \qquad \gamma(t) = \theta^{(p)}(t + s) .
\]
The domain \(\mathcal{D}^{(p)} - s\) is an open interval containing \(0\) (it is the
translate of an open interval containing \(s\) by \(-s\)). The
translation lemma
asserts that \(\gamma\) is an integral curve of \(V\), and by construction
\(\gamma(0) = \theta^{(p)}(s) = q\), so \(\gamma\) is an integral curve of \(V\)
starting at \(q\). By the maximality of \(\theta^{(q)}\), the domain of \(\gamma\) is
contained in \(\mathcal{D}^{(q)}\), and the Step 1 uniqueness lemma forces
\(\gamma(t) = \theta^{(q)}(t)\) on that domain. The inclusion of domains says
\[
\mathcal{D}^{(p)} - s \subseteq \mathcal{D}^{(q)} ,
\]
and the equality of values says that on this common interval,
\[
\theta^{(q)}(t) = \theta^{(p)}(t + s) ,
\]
which is the group law of the maximal flow.
For the reverse inclusion, apply the same construction to the pair \((-s, q)\) in
place of \((s, p)\): noting that \(-s \in \mathcal{D}^{(q)}\) and
\(\theta^{(q)}(-s) = p\) by the group law just established, the same argument yields
\(\mathcal{D}^{(q)} - (-s) \subseteq \mathcal{D}^{(p)}\), which is to say
\(\mathcal{D}^{(q)} + s \subseteq \mathcal{D}^{(p)}\), or equivalently
\(\mathcal{D}^{(q)} \subseteq \mathcal{D}^{(p)} - s\). Combining the two inclusions
gives \(\mathcal{D}^{(q)} = \mathcal{D}^{(p)} - s\), which is property (b).
With (a) and (b) in hand, the maximal flow as a set-theoretic object is fully
constructed: take
\[
\mathcal{D} = \{ (t, p) \in \mathbb{R} \times M : t \in \mathcal{D}^{(p)} \} ,
\]
and define \(\theta : \mathcal{D} \to M\) by \(\theta(t, p) = \theta^{(p)}(t)\). The
identity \(\theta(0, p) = p\) holds by the definition of an integral curve, and the
group law \(\theta(t, \theta(s, p)) = \theta(t + s, p)\) is exactly the equality
\(\theta^{(q)}(t) = \theta^{(p)}(t + s)\) just derived, valid wherever both sides are
defined. What remains to verify is that \(\mathcal{D}\) is open in
\(\mathbb{R} \times M\) and that \(\theta\) is smooth on it — both of which are
established in the next step.
Step 4: The Flow Domain is Open and the Flow is Smooth
The Step 2 and Step 3 constructions yield a map \(\theta : \mathcal{D} \to M\) and a
set \(\mathcal{D} \subseteq \mathbb{R} \times M\) satisfying the algebraic properties
required of a flow. To complete the proof of existence, we must show that
\(\mathcal{D}\) is open in \(\mathbb{R} \times M\) — so that it is a flow domain in the
sense of the definition above — and that \(\theta\) is smooth as a map between
manifolds.
The strategy is to identify the largest subset of \(\mathcal{D}\) on which the desired
smoothness is already known, and to use the group law to propagate that smoothness
along trajectories until it fills \(\mathcal{D}\). Let \(W \subseteq \mathcal{D}\)
consist of all points \((t, p) \in \mathcal{D}\) such that \(\theta\) is defined and
smooth on some product neighborhood of \((t, p)\) of the form \(J \times U \subseteq
\mathcal{D}\), where \(J\) is an open interval containing both \(0\) and \(t\) and
\(U\) is a neighborhood of \(p\). By construction \(W\) is open in \(\mathbb{R} \times M\),
and \(\theta\) is smooth on \(W\). We claim \(W = \mathcal{D}\).
The smooth-dependence statement for solutions of smooth autonomous systems of ordinary
differential equations — the companion of the existence and uniqueness statements
invoked earlier, with the same underlying analytical origin — produces a product
neighborhood of \((0, p)\) inside \(\mathcal{D}\) on which \(\theta\) is defined and
smooth, for every \(p \in M\). The slice \(\{0\} \times M\) is therefore contained in
\(W\). Suppose for contradiction that \(W \neq \mathcal{D}\), and pick
\((\tau, p_0) \in \mathcal{D} \setminus W\); by considering \(-\tau\) and the flow of
\(-V\) if necessary, we may assume \(\tau > 0\). Let
\[
t_0 = \inf \{ t \in (0, \tau] : (t, p_0) \notin W \} ,
\]
the smallest positive time at which the smoothness produced by the local statement
fails to extend along the trajectory through \(p_0\). Since \(\{0\} \times M \subseteq W\),
one has \(t_0 > 0\), and \((t_0, p_0)\) lies on the boundary of \(W\) in the closed
subinterval \([0, \tau] \times \{p_0\}\).
The contradiction is engineered by extending the smoothness across \(t_0\). Choose
\(t_1 < t_0\) close enough to \(t_0\) that \(\theta(t_1, p_0)\) lies in a neighborhood
where the local smoothness statement applies, and a positive \(\varepsilon\) with
\(t_1 + \varepsilon > t_0\). Because \((t_1, p_0) \in W\), there is a product
neighborhood \((t_1 - \delta, t_1 + \delta) \times U_1 \subseteq W\) on which
\(\theta\) is smooth, and \(U_1\) may be shrunk so that \(\theta(\{t_1\} \times U_1)\)
lies in the local-smoothness neighborhood of \(\theta(t_1, p_0)\). The group law
\(\theta(t, p) = \theta\bigl( t - t_1, \, \theta(t_1, p) \bigr)\) then expresses
\(\theta\) on a product neighborhood \([0, t_1 + \varepsilon) \times U_1\) as a
composition of two maps each known to be smooth on the relevant domain — the inner
map \((t, p) \mapsto \theta(t_1, p)\) is smooth on \(\{t_1\} \times U_1\) by the
inclusion of this product in \(W\), and the outer map
\((s, q) \mapsto \theta(s, q)\) is smooth on the product
\([0, \varepsilon) \times \theta(\{t_1\} \times U_1)\) by the local-smoothness
statement at \(\theta(t_1, p_0)\). Hence \(\theta\) is defined and smooth on an open
neighborhood of \((t_0, p_0)\), contradicting the choice of \(t_0\). Thus
\(W = \mathcal{D}\), the set \(\mathcal{D}\) is open, and \(\theta\) is smooth on it.
Step 5: The Time-\(t\) Maps Are Diffeomorphisms
Property (c) is now an immediate corollary of (b). Fix \(t \in \mathbb{R}\). The
equivalence \(p \in M_t \iff t \in \mathcal{D}^{(p)}\) and property (b) give
\[
\begin{align*}
p \in M_t
&\iff t \in \mathcal{D}^{(p)} \\\\
&\iff -t \in \mathcal{D}^{(p)} - t = \mathcal{D}^{(\theta(t, p))} \\\\
&\iff \theta(t, p) \in M_{-t} ,
\end{align*}
\]
so \(\theta_t\) is a map \(M_t \to M_{-t}\). The openness of \(M_t\) is the assertion
that the slice of \(\mathcal{D}\) at height \(t\) is open in \(M\), which follows from
Step 4. Applying the same reasoning with \(-t\) in place of \(t\) and using the group
laws \(\theta_{-t} \circ \theta_t = \theta_0 = \mathrm{Id}\) on \(M_t\) and
\(\theta_t \circ \theta_{-t} = \mathrm{Id}\) on \(M_{-t}\) shows that
\(\theta_t : M_t \to M_{-t}\) is a bijection with two-sided inverse \(\theta_{-t}\).
Both maps are smooth as restrictions of the smooth map \(\theta\) of Step 4 to slices
of \(\mathcal{D}\), so each is a diffeomorphism.
The uniqueness of the flow follows from the uniqueness of its trajectories. If
\(\tilde\theta : \tilde{\mathcal{D}} \to M\) is another smooth maximal flow with
infinitesimal generator \(V\), then for each \(p \in M\) the trajectory
\(\tilde\theta^{(p)}\) is an integral curve of \(V\) starting at \(p\). By maximality,
\(\tilde{\mathcal{D}}^{(p)} \subseteq \mathcal{D}^{(p)}\), and by the Step 1
uniqueness lemma the values agree on \(\tilde{\mathcal{D}}^{(p)}\). Maximality of
\(\tilde\theta\) forces equality of domains, hence \(\tilde\theta = \theta\). This
completes the proof of the theorem.
The flow whose existence and uniqueness the theorem asserts is called the
flow generated by \(V\), or simply the flow of
\(V\). The name "infinitesimal generator" reflects the picture the theorem makes
rigorous: in a smooth chart, a good approximation to an integral curve of \(V\) on a
short time interval is the straight-line motion in the direction of \(V_p\) starting
at \(p\), and the full integral curve is built up by composing many such infinitesimal
motions, with the direction of each motion determined by the value of \(V\) at the
point arrived at in the previous step. The vector field at \(p\) sets the initial
velocity; the field along the trajectory determines its continuation.
The Flow of an Autonomous Vector Field, and the Limits of That Setting
Constructions in machine learning that build models of continuous-time dynamics
on top of flows of vector fields rely on the well-posedness of an ordinary
differential equation \(\dot{\mathbf{x}} = \mathbf{u}(\mathbf{x})\) — that
distinct trajectories do not cross, that solutions depend smoothly on initial
conditions, and that the time-\(t\) map is a diffeomorphism. The
flow-matching construction
in generative modeling, for instance, invokes the Picard–Lindelöf
theorem as a forward-declared regularity input under a Lipschitz assumption.
Under the smoothness hypothesis on \(V\), the present theorem establishes the
autonomous, manifold-level realization of that input.
What is not settled here is the case actually used in such constructions,
where the velocity field \(\mathbf{u}_t(\mathbf{x})\) depends nontrivially on
the time parameter \(t\). A time-dependent velocity field is not a vector field
on the ambient space alone but a smooth map \(\mathbb{R} \times M \to TM\)
covering the projection onto the second factor, and its well-posedness theory
parallels but does not coincide with the autonomous one. A future development of
this series will establish the corresponding theorem for time-dependent smooth
vector fields; the present theorem covers the autonomous regime only.
Naturality and Diffeomorphism Invariance
The flow construction associates to each smooth vector field on a smooth manifold a
canonical maximal flow. The next question — typical for any construction with a claim
to being canonical — is how this association interacts with smooth maps between
manifolds. The
naturality of integral curves
asserted that, for an \(F\)-related pair of vector fields, \(F\) sends integral curves
of one to integral curves of the other. Translated into the language of flows, this
becomes a commutation relation between the flows themselves.
Proposition (Naturality of Flows)
Let \(F : M \to N\) be a smooth map between smooth manifolds, \(X \in \mathfrak{X}(M)\),
\(Y \in \mathfrak{X}(N)\), and suppose \(X\) and \(Y\) are
\(F\)-related.
Let \(\theta : \mathcal{D} \to M\) be the flow of \(X\), and \(\eta : \mathcal{E} \to N\)
the flow of \(Y\). Then for each \(t \in \mathbb{R}\),
\[
F(M_t) \subseteq N_t , \qquad \eta_t \circ F = F \circ \theta_t \quad \text{on } M_t .
\]
Proof:
Fix \(p \in M\) and let \(\theta^{(p)} : \mathcal{D}^{(p)} \to M\) be the maximal
integral curve of \(X\) starting at \(p\). The naturality of integral curves applied
to \(\theta^{(p)}\) shows that \(F \circ \theta^{(p)} : \mathcal{D}^{(p)} \to N\) is
an integral curve of \(Y\), starting at the point \(F(\theta^{(p)}(0)) = F(p)\). The
maximal integral curve of \(Y\) starting at \(F(p)\) is \(\eta^{(F(p))}\), defined
on the maximal interval \(\mathcal{E}^{(F(p))}\). Maximality of
\(\eta^{(F(p))}\) forces the domain inclusion
\(\mathcal{D}^{(p)} \subseteq \mathcal{E}^{(F(p))}\); the uniqueness lemma of
Step 1 of the fundamental theorem then identifies the two integral curves of \(Y\)
that pass through \(F(p)\) at time \(0\) on this common interval,
\[
F \circ \theta^{(p)} = \eta^{(F(p))} \qquad \text{on } \mathcal{D}^{(p)} .
\]
The inclusion \(F(M_t) \subseteq N_t\) and the commutation relation now follow by
bookkeeping. For \(p \in M_t\) we have \(t \in \mathcal{D}^{(p)}\), hence
\(t \in \mathcal{E}^{(F(p))}\) by the just-established containment, hence
\(F(p) \in N_t\). Evaluating the equality of trajectories at \(t\) gives
\(F(\theta_t(p)) = \eta_t(F(p))\), which is the commutation relation.
The content of the proposition is captured by the commutative diagram of smooth maps
\[
\begin{array}{ccc}
M_t & \xrightarrow{F} & N_t \\\\
{\scriptstyle \theta_t} \downarrow & & \downarrow {\scriptstyle \eta_t} \\\\
M_{-t} & \xrightarrow{F} & N_{-t} .
\end{array}
\]
The horizontal arrows are restrictions of \(F\); the vertical arrows are the
time-\(t\) maps of the two flows, which are diffeomorphisms by part (c) of the
fundamental theorem. The diagram says exactly that following \(X\) for time \(t\) and
then applying \(F\) gives the same result as first applying \(F\) and then following
\(Y\) for time \(t\) — the operation of "flow for time \(t\)" commutes with \(F\)
whenever \(X\) and \(Y\) are \(F\)-related.
The special case in which \(F\) is itself a diffeomorphism deserves a separate
statement, because in that case the \(F\)-relatedness condition can be packaged as a
pushforward of vector fields.
Corollary (Diffeomorphism Invariance of Flows)
Let \(F : M \to N\) be a diffeomorphism between smooth manifolds and let
\(X \in \mathfrak{X}(M)\). If \(\theta\) is the flow of \(X\) on the flow domain
\(\mathcal{D} \subseteq \mathbb{R} \times M\), then the flow of the
pushforward vector field
\(F_* X \in \mathfrak{X}(N)\) is the map \(\eta : F(\mathcal{D}) \to N\) defined by
\[
\eta_t = F \circ \theta_t \circ F^{-1} ,
\]
with \(N_t = F(M_t)\) for each \(t \in \mathbb{R}\).
Proof:
Because \(F\) is a diffeomorphism, \(X\) and \(F_* X\) are \(F\)-related by the
defining property of the pushforward. The previous proposition therefore gives
\(F(M_t) \subseteq N_t\) and \(\eta_t \circ F = F \circ \theta_t\) on \(M_t\),
where \(\eta\) is the maximal flow of \(F_* X\) on its own flow domain. Applying
the same proposition to \(F^{-1}\), which pulls \(F_* X\) back to \(X\), yields
\(F^{-1}(N_t) \subseteq M_t\) and \(\theta_t \circ F^{-1} = F^{-1} \circ \eta_t\)
on \(N_t\). The two inclusions combine to give \(F(M_t) = N_t\), and the
commutation relation rearranges to \(\eta_t = F \circ \theta_t \circ F^{-1}\).
Complete Vector Fields
The counterexamples of the first section showed that not every smooth vector field
generates a global flow: integral curves can run off the manifold in finite time, or
blow up to infinity along a coordinate direction. The vector fields that do generate
global flows form a distinguished class, and many structural statements about flows
on a manifold depend on identifying when a given field belongs to that class.
Definition: Complete Vector Field
A smooth vector field \(V\) on a smooth manifold \(M\) is complete
if its flow is a
global flow:
equivalently, if the maximal flow domain of \(V\) is all of
\(\mathbb{R} \times M\), or equivalently, if every maximal integral curve of \(V\)
is defined on the whole real line.
The vector fields whose global flows were written down explicitly earlier are
complete; the two counterexamples of the first section, in contrast, are incomplete
— one because the manifold loses a point along the trajectory, the other because the
coordinate escapes to infinity in finite time. The general question is asymmetric:
to show that a vector field is incomplete, a single maximal integral curve with
bounded domain suffices, typically produced by solving the equation in closed form;
to show that a vector field is complete, one must control every maximal integral
curve, which in general cannot be done by explicit integration. The remainder of
this section develops criteria that extract completeness from structural features of
the manifold or of the vector field — compact support, or invariance under a group
of translations — without solving the differential equation.
A Uniform Lower Bound on Existence Times
The first criterion records a useful general principle: if there is a single
\(\varepsilon > 0\) that lower-bounds the existence interval of every trajectory, then
every trajectory in fact exists for all time. The bound need not be tight, and the
\(\varepsilon\) does not appear in the conclusion; what matters is that one such
\(\varepsilon\) exists, uniformly across the manifold.
Proof:
Suppose for contradiction that \(V\) is not complete. Then there is some
\(p \in M\) for which the maximal existence interval \(\mathcal{D}^{(p)}\) is a
proper subinterval of \(\mathbb{R}\). Assume \(\mathcal{D}^{(p)}\) is bounded
above; the case in which it is bounded below is handled by the same argument
applied to the time-reversed flow. Let \(b = \sup \mathcal{D}^{(p)} < +\infty\),
and choose \(t_0 \in \mathcal{D}^{(p)}\) with
\[
b - \varepsilon < t_0 < b .
\]
Set \(q = \theta^{(p)}(t_0) \in M\).
The hypothesis applied at \(q\) gives
\(\mathcal{D}^{(q)} \supseteq (-\varepsilon, \varepsilon)\), so the trajectory
\(\theta^{(q)}\) is defined on this open interval. Define a curve
\(\gamma : \mathcal{D}^{(p)} \cup (t_0 - \varepsilon, t_0 + \varepsilon) \to M\)
by patching two pieces:
\[
\gamma(t) =
\begin{cases}
\theta^{(p)}(t) , & t \in \mathcal{D}^{(p)} , \\\\
\theta^{(q)}(t - t_0) , & t \in (t_0 - \varepsilon, \, t_0 + \varepsilon) .
\end{cases}
\]
The two pieces overlap on the open interval
\((t_0 - \varepsilon, \, b)\), which is contained in \(\mathcal{D}^{(p)}\): the
uniform-time hypothesis applied at \(p\) gives
\(\mathcal{D}^{(p)} \supseteq (-\varepsilon, \varepsilon)\), hence \(b \geq
\varepsilon\), and the choice \(t_0 > b - \varepsilon\) together with
\(\inf \mathcal{D}^{(p)} \leq -\varepsilon\) forces \(t_0 - \varepsilon >
\inf \mathcal{D}^{(p)}\), so \((t_0 - \varepsilon, b) \subseteq \mathcal{D}^{(p)}\).
On this overlap, the
translation lemma
identifies \(t \mapsto \theta^{(q)}(t - t_0)\) as an integral curve of \(V\)
starting at \(q = \theta^{(p)}(t_0)\) when \(t = t_0\); the same is true of
\(\theta^{(p)}\) on \(\mathcal{D}^{(p)}\), restricted near \(t_0\). The
uniqueness of integral curves established earlier therefore forces the two pieces
to coincide on the overlap, so \(\gamma\) is well-defined.
The curve \(\gamma\) is an integral curve of \(V\) on its open domain: on each of
the two pieces it is, and the pieces glue smoothly along the overlap. Its starting
point is \(\gamma(0) = \theta^{(p)}(0) = p\), since \(0 \in \mathcal{D}^{(p)}\) by
the definition of a flow domain. The domain of \(\gamma\) is the union
\(\mathcal{D}^{(p)} \cup (t_0 - \varepsilon, t_0 + \varepsilon)\), which by the
overlap inclusion just established equals
\((\inf \mathcal{D}^{(p)}, \, t_0 + \varepsilon)\); the choice of \(t_0\)
guarantees \(t_0 + \varepsilon > b\), so this interval strictly contains
\(\mathcal{D}^{(p)} = (\inf \mathcal{D}^{(p)}, b)\). Thus \(\gamma\) is an
integral curve of \(V\) starting at \(p\) and defined on an open interval
strictly containing \(\mathcal{D}^{(p)}\). This contradicts the maximality of
\(\theta^{(p)}\). The contradiction shows \(\mathcal{D}^{(p)}\) is not bounded
above; the symmetric argument shows it is not bounded below; hence
\(\mathcal{D}^{(p)} = \mathbb{R}\) for every \(p \in M\), and \(V\) is complete.
The uniform time lemma reduces the question of completeness to the question of a
single uniform bound on existence intervals. In the applications that follow, this
bound is produced by very different mechanisms — by compactness of the support of
the vector field, by compactness of the underlying manifold, or by group-theoretic
homogeneity — but in each case the lemma serves as the universal final step that
turns local-in-time information into a global flow.
Compact Support and Compact Manifolds
The first geometric application of the uniform time lemma applies whenever the region
on which the vector field is nonzero is compact. Outside that region the trajectories
are constant, and inside it a single \(\varepsilon\) can be extracted from a finite
cover of local flow domains.
Theorem (Compactly Supported Vector Fields Are Complete)
Every compactly supported smooth vector field on a smooth manifold is complete.
Proof:
Let \(V\) be a smooth vector field on \(M\) with
compact support
\(K = \mathrm{supp}\, V\). For each \(p \in K\), the fundamental theorem provides
an open neighborhood of \((0, p)\) inside the flow domain \(\mathcal{D}\) on
which \(\theta\) is defined and smooth; such a neighborhood may be taken of
product form \((-\varepsilon_p, \varepsilon_p) \times U_p\), where
\(\varepsilon_p > 0\) and \(U_p\) is an open neighborhood of \(p\). The
collection \(\{ U_p \}_{p \in K}\) is an open cover of \(K\) by open subsets of
\(M\). By the
defining property of compactness,
there is a finite subcover \(U_{p_1}, \ldots, U_{p_n}\) of \(K\); set
\[
\varepsilon = \min \{ \varepsilon_{p_1}, \ldots, \varepsilon_{p_n} \} > 0 .
\]
We verify that \(\mathcal{D}^{(q)} \supseteq (-\varepsilon, \varepsilon)\) for
every \(q \in M\), in two cases.
Case 1: \(q \in K\). Then \(q \in U_{p_i}\) for some \(i\), and the
product neighborhood \((-\varepsilon_{p_i}, \varepsilon_{p_i}) \times U_{p_i}\)
is contained in \(\mathcal{D}\) by construction. In particular
\(\{ t \in \mathbb{R} : (t, q) \in \mathcal{D} \} \supseteq (-\varepsilon_{p_i}, \varepsilon_{p_i}) \supseteq (-\varepsilon, \varepsilon)\),
which is the inclusion claimed.
Case 2: \(q \notin K\). Then \(V\) vanishes on an open neighborhood of
\(q\) (the complement of \(K\) is open, and \(K\) contains the closure of the
set on which \(V \neq 0\)). The constant curve \(\gamma : \mathbb{R} \to M\) with
\(\gamma(t) \equiv q\) has velocity zero everywhere, so
\(\gamma'(t) = 0 = V_{\gamma(t)}\) for all \(t\); hence \(\gamma\) is an integral
curve of \(V\) defined on all of \(\mathbb{R}\), and the maximal integral curve
through \(q\) has \(\mathcal{D}^{(q)} = \mathbb{R}\), which certainly contains
\((-\varepsilon, \varepsilon)\).
The hypothesis of the uniform time lemma is therefore satisfied, and \(V\) is
complete.
On a compact manifold every smooth vector field has compact support — its support is
a closed subset of a compact space — so the theorem specializes to a striking
statement: compactness of the underlying manifold by itself forces completeness of
every smooth dynamical system one can write down.
Corollary (Completeness on Compact Manifolds)
Every smooth vector field on a compact smooth manifold is complete.
Proof:
Let \(M\) be a compact smooth manifold and \(V \in \mathfrak{X}(M)\). The support
\(\mathrm{supp}\, V\) is a closed subset of \(M\), and every closed subset of a
compact space is compact, so \(V\) is compactly supported. The previous theorem
applies.
Compactness of the manifold eliminates by hand both modes of incompleteness
encountered in the counterexamples — the manifold is too large to be exited in
finite time, and there is no infinity toward which a coordinate could escape.
Left-Invariant Vector Fields on Lie Groups
The second geometric application of the uniform time lemma replaces compactness by a
different structural hypothesis: invariance under a transitive group of
diffeomorphisms of the manifold. On a Lie group, the left translations supply such a
group, and the corresponding distinguished class of vector fields is the
left-invariant ones. The conclusion is that every such field — without any compactness
or boundedness assumption on the group — is complete.
Proof:
Let \(G\) be a Lie group and \(X\) a left-invariant smooth vector field on \(G\),
with maximal flow \(\theta : \mathcal{D} \to G\). The local existence statement of
the earlier development applied at the identity element \(e \in G\) yields some
\(\varepsilon > 0\) and a smooth integral curve
\(\theta^{(e)} : (-\varepsilon, \varepsilon) \to G\) of \(X\) starting at \(e\).
We propagate this \(\varepsilon\) to every other point of \(G\) by left
translation.
Fix \(g \in G\) and let
\(L_g : G \to G\)
be left translation by \(g\). The defining property of a left-invariant vector
field is that \(X\) and itself are \(L_g\)-related — that is, \(X\) is
\(L_g\)-related to itself for every \(g \in G\). The
naturality of integral curves
applied to \(L_g\), \(X\), and \(X\) itself shows that
\(L_g \circ \theta^{(e)} : (-\varepsilon, \varepsilon) \to G\) is an integral
curve of \(X\); its starting value is
\[
L_g \bigl( \theta^{(e)}(0) \bigr) = L_g(e) = g \cdot e = g .
\]
Thus \(L_g \circ \theta^{(e)}\) is an integral curve of \(X\) starting at \(g\),
defined on the open interval \((-\varepsilon, \varepsilon)\).
By the maximality of \(\theta^{(g)}\), the domain of any integral curve of \(X\)
starting at \(g\) is contained in \(\mathcal{D}^{(g)}\); in particular,
\[
(-\varepsilon, \varepsilon) \subseteq \mathcal{D}^{(g)} .
\]
The element \(g \in G\) was arbitrary, so the inclusion holds for every \(g\),
with the same \(\varepsilon\) throughout. The hypothesis of the
uniform time lemma
is therefore satisfied, and \(X\) is complete.
One-Parameter Subgroups, Revisited
The earlier development of matrix Lie groups established that every
one-parameter subgroup
of a matrix Lie group has the form \(\gamma(t) = \exp(tA)\) for a unique matrix
\(A = \gamma'(0)\), defined for every \(t \in \mathbb{R}\). The completeness
statement just proved is the corresponding result in the abstract setting: for
any Lie group \(G\) and any left-invariant smooth vector field \(X\), the
trajectory \(\theta^{(e)} : \mathbb{R} \to G\) through the identity is a smooth
group homomorphism — a one-parameter subgroup of \(G\) — and conversely every
smooth one-parameter subgroup arises this way. The
matrix-exponential identification
is the matrix-group instance of an abstract object whose existence rests on the
present completeness theorem.
This is the construction known in the general case as the exponential map of
the Lie group: a tangent vector at the identity determines a left-invariant
vector field, whose flow is global by the theorem above, and whose time-\(1\)
value gives back the exponential of the original tangent vector. The
differential at the origin and the compatibility with the bracket on the Lie
algebra belong to later stages of the development; the existence and global
character of the underlying one-parameter subgroups have now been settled in
full generality.