Flowouts & Canonical Form Theorem

Suppose for contradiction that there exists \(t_0 \in J\) and a compact subset \(K \subseteq M\) with \(\gamma\bigl([t_0, b)\bigr) \subseteq K\). Choose any sequence \(t_k \in [t_0, b)\) with \(t_k \to b\). The points \(\gamma(t_k)\) lie in the compact set \(K\), so a subsequence, still denoted \(\gamma(t_k)\) for simplicity, converges to some point \(q \in K\).

Applying the local existence theorem for integral curves at \(q\), we obtain an integral curve \(\sigma : (-\varepsilon, \varepsilon) \to M\) of \(V\) with \(\sigma(0) = q\) for some \(\varepsilon \gt 0\). The point \((0, q)\) lies in \(\mathcal{D}\) because \(\mathcal{D}\) contains \(\{0\} \times M\). The openness of the flow domain \(\mathcal{D}\) at that point gives an open product neighborhood \((-\eta, \eta) \times U \subseteq \mathcal{D}\) for some \(\eta \in (0, \varepsilon)\) and some open neighborhood \(U\) of \(q\) in \(M\). This is the geometric content of smooth dependence on initial conditions. For every \(r \in U\), the maximal interval \(\mathcal{D}^{(r)}\) contains \((-\eta, \eta)\), so all integral curves starting in \(U\) are simultaneously defined on the interval \((-\eta, \eta)\).

Fix \(k\) large enough that \(\gamma(t_k) \in U\) and \(b - t_k \lt \eta\), and write \(q_k = \gamma(t_k)\). Let \(\sigma_k : (-\eta, \eta) \to M\) be the restriction to \((-\eta, \eta)\) of the trajectory \(\theta^{(q_k)}\), which is defined on that interval by the previous paragraph. The translation lemma identifies the curve \(\tilde\gamma : (t_k - \eta, \, t_k + \eta) \to M\) defined by \(\tilde\gamma(t) = \sigma_k(t - t_k)\) as an integral curve of \(V\). Setting \(t = t_k\) gives \(\tilde\gamma(t_k) = \sigma_k(0) = q_k\). The same lemma turns \(\gamma\) into the integral curve \(t \mapsto \gamma(t + t_k)\) on \(J - t_k\), which starts at \(q_k\) and is again maximal, since an extension of it would translate back to an extension of \(\gamma\). The uniqueness clause of the fundamental theorem identifies this translate with \(\theta^{(q_k)}\). Evaluating both descriptions at \(t - t_k\) gives \(\gamma(t) = \sigma_k(t - t_k) = \tilde\gamma(t)\) for every \(t\) in the overlap \(J \cap (t_k - \eta, t_k + \eta)\).

Define a curve \(\hat\gamma\) on the open interval \(J \cup (t_k - \eta, \, t_k + \eta)\) by taking \(\gamma\) on \(J\) and \(\tilde\gamma\) on \((t_k - \eta, t_k + \eta)\). The two pieces agree on the overlap by the previous paragraph, so \(\hat\gamma\) is well-defined, and each piece is an integral curve of \(V\) on its own domain. Hence \(\hat\gamma\) is an integral curve of \(V\) on the union. The choice \(b - t_k \lt \eta\) makes \(t_k + \eta \gt b\), so the union strictly contains \(J\), and \(\hat\gamma\) is an integral curve of \(V\) extending \(\gamma\) to a strictly larger open interval. This contradicts the maximality of \(\gamma\), so the assumption that \(\gamma\bigl([t_0, b)\bigr)\) lies in a compact set is untenable.

Fix \(p \in S\), and let \(\sigma : \mathcal{D}^{(p)} \to \mathbb{R} \times S\) be the curve \(\sigma(t) = (t, p)\). Its image lies in \(\mathcal{O}\) by the definition of \(\mathcal{O}\). The composition \(\Phi \circ \sigma : t \mapsto \theta(t, p) = \theta^{(p)}(t)\) is the maximal integral curve of \(V\) through \(p\), so for every \(t_0 \in \mathcal{D}^{(p)}\), \[ d\Phi_{(t_0, p)}\!\left( \frac{\partial}{\partial t}\bigg|_{(t_0, p)} \right) = (\Phi \circ \sigma)'(t_0) = V_{\Phi(t_0, p)} . \] The first equality is the chain rule, and the second is the integral-curve equation \(\theta^{(p)\prime}(t_0) = V_{\theta(t_0, p)}\). The identity is exactly the condition that \(\partial/\partial t\) and \(V\) are \(\Phi\)-related.

We first check that \(d\Phi_{(0, p)}\) is injective for every \(p \in S\), then propagate the conclusion to general points \((t_0, p_0) \in \mathcal{O}\) by a translation argument.

At the point \((0, p)\), the restriction of \(\Phi\) to the slice \(\{0\} \times S \subseteq \mathcal{O}\) is the composition of the canonical identification \(\{0\} \times S \cong S\) with the inclusion \(S \hookrightarrow M\). Both are embeddings, so the restriction \(\Phi|_{\{0\} \times S}\) is an embedding too. The differential \(d\Phi_{(0, p)}\) therefore carries the subspace \(T_{(0, p)}(\{0\} \times S) \cong T_p S\) injectively into \(T_p M\), and in fact takes it to the subspace \(T_p S \subseteq T_p M\) by the inclusion identification.

Pick a basis \((E_1, \ldots, E_k)\) of \(T_p S\). Then \(\bigl(\partial/\partial t|_{(0, p)}, E_1, \ldots, E_k\bigr)\) is a basis of \(T_{(0, p)}\mathcal{O} \cong T_0\mathbb{R} \oplus T_p S\). Applying \(d\Phi_{(0, p)}\) and using the formula just derived together with the inclusion of \(T_p S\) into \(T_p M\), we find that this basis maps to \((V_p, E_1, \ldots, E_k)\) inside \(T_p M\). The hypothesis that \(V\) is nowhere tangent to \(S\) gives \(V_p \notin T_p S\). The resulting \((k + 1)\)-tuple is therefore linearly independent, and \(d\Phi_{(0, p)}\) is injective.

Now fix an arbitrary \((t_0, p_0) \in \mathcal{O}\). The translation map \(\tau_{t_0} : (t, p) \mapsto (t + t_0, p)\) is defined on the open set \(\mathcal{O} - (t_0, 0) = \{(t, p) : (t + t_0, p) \in \mathcal{O}\}\). This set contains \((0, p_0)\) because \((t_0, p_0) \in \mathcal{O}\). Shrinking to a small open neighborhood of \((0, p_0)\) inside it if necessary, we make both \(\tau_{t_0}\) and its inverse \(\tau_{-t_0}\) smooth and bijective onto their images in \(\mathcal{O}\). The time-\(t_0\) flow map \(\theta_{t_0}\), defined on the open set \(M_{t_0} = \{p \in M : (t_0, p) \in \mathcal{D}\}\), is a diffeomorphism onto its image by the fundamental theorem on flows. On a neighborhood of \((0, p_0)\) where all four maps below are simultaneously defined, the flow group law gives the commutative diagram \[ \begin{array}{ccc} \mathcal{O} & \xrightarrow{\tau_{t_0}} & \mathcal{O} \\\\ \Phi \downarrow & & \downarrow \Phi \\\\ M & \xrightarrow{\theta_{t_0}} & M , \end{array} \] the verification being \(\theta_{t_0}(\Phi(t, p)) = \theta_{t_0}(\theta(t, p)) = \theta(t + t_0, p) = \Phi(t + t_0, p) = \Phi(\tau_{t_0}(t, p))\). Taking differentials at \((0, p_0)\), we find that the horizontal differentials are isomorphisms, so the two vertical maps \(d\Phi_{(0, p_0)}\) and \(d\Phi_{(t_0, p_0)}\) have the same rank. We have already shown that \(d\Phi_{(0, p_0)}\) is injective, so \(d\Phi_{(t_0, p_0)}\) is too. This is the immersion property.

Fix \(p_0 \in S\). Since \(S\) is embedded, the local slice criterion supplies a slice chart \((U, (x^i))\) for \(S\) in \(M\) centered at \(p_0\), so that \(U \cap S = \{x^{k+1} = \cdots = x^n = 0\}\) where \(n = \dim M\). Because \(V\) is not tangent to \(S\), some component \(V^j(p_0)\) with \(j \gt k\) is nonzero. By continuity, \(V^j\) does not vanish on a neighborhood of \(p_0\) and therefore retains a single sign there. Shrinking \(U\) to such a neighborhood, we obtain a constant \(c \gt 0\) such that \[ V^j(p) \geq c \quad \text{or} \quad V^j(p) \leq -c \quad \text{for all } p \in U , \] and the sign is the same as the sign of \(V^j(p_0)\) throughout \(U\). The component \(\Phi^j(t, p)\) of \(\Phi\) in this chart satisfies \(\partial \Phi^j / \partial t = V^j \circ \Phi\), which is the coordinate reading of the \(\Phi\)-relation in (b). The same component satisfies \(\Phi^j(0, p) = 0\) for \(p \in S\), because \(\Phi(0, p) = p\) lies on the slice. Choose a product neighborhood \((-\varepsilon_{p_0}, \varepsilon_{p_0}) \times W_{p_0}\) of \((0, p_0)\) in \(\mathcal{O}\) so that \(\Phi\) maps it into \(U\). The sign-preserving lower bound for \(V^j\) on \(U\) gives a corresponding lower bound for \(\partial \Phi^j / \partial t\) on the product neighborhood, and the fundamental theorem of calculus yields \[ \begin{align*} \Phi^j(t, p) &= \int_0^t \frac{\partial \Phi^j}{\partial s}(s, p) \, ds \\\\ &= \int_0^t V^j\bigl( \Phi(s, p) \bigr) \, ds. \end{align*} \] The integrand has constant sign and absolute value bounded below by \(c\), so \(|\Phi^j(t, p)| \geq c |t|\) on the product neighborhood. The \(j\)-th coordinate vanishes on \(S \cap U\), so this lower bound implies \(\Phi(t, p) \in S \iff t = 0\) on the product neighborhood. For a second application, suppose \(\Phi(t, p) = \Phi(t', p)\) for the same \(p \in W_{p_0}\) and \(t, t' \in (-\varepsilon_{p_0}, \varepsilon_{p_0})\). Then \(t \mapsto \Phi(t, p)\) is the integral curve \(\theta^{(p)}\) of \(V\) restricted to a small interval on which \(\Phi^j\) is strictly monotone in \(t\). A strictly monotone function is injective, so \(t = t'\).

These constructions give, for each \(p \in S\), an open neighborhood \(W_p \subseteq S\) and a positive number \(\varepsilon_p\) such that, on \((-\varepsilon_p, \varepsilon_p) \times W_p\), the map \(\Phi\) takes values in \(S\) only at \(t = 0\) and is injective in \(t\) for each fixed point of \(W_p\). The cover \(\{W_p : p \in S\}\) of \(S\) admits a smooth partition of unity \(\{\psi_p\}\) subordinate to it, and the function \[ f(q) = \sum_{p \in S} \varepsilon_p \psi_p(q) \] is smooth and positive on \(S\). The defining property of the partition of unity gives a useful bound. For each \(q \in S\), the values \(\psi_p(q)\) sum to one, and the sum involves only finitely many \(p\) with \(\psi_p(q) \gt 0\). Choosing among these the index \(p_0\) with the largest \(\varepsilon_{p_0}\) yields \(f(q) \leq \varepsilon_{p_0}\).

Set \(\delta = f / 2\) and suppose \(\Phi(t, q) = \Phi(t', q')\) for two points \((t, q)\) and \((t', q')\) of \(\mathcal{O}_\delta\). Without loss of generality \(f(q') \leq f(q)\). Since the values \(\psi_p(q)\) sum to one in the partition of unity, choose an index \(p_0\) with \(\psi_{p_0}(q) \gt 0\) and \(\varepsilon_{p_0}\) maximal among such indices. Then \(q \in W_{p_0}\) by the support condition on \(\psi_{p_0}\), and \(f(q) \leq \varepsilon_{p_0}\) by the bound established above. Apply the flow group law to the equation \(\theta_t(q) = \theta_{t'}(q')\). On the side of \(q\), this rewrites as \(\theta_{-t'}(\theta_t(q)) = q'\), and the group law identifies the left side with \(\theta_{t - t'}(q) = \Phi(t - t', q)\). Therefore \[ \Phi(t - t', q) = q' \in S. \] The triangle inequality combined with \(|t| \lt \delta(q) = f(q) / 2\) and \(|t'| \lt \delta(q') = f(q') / 2 \leq f(q) / 2\) gives \[ |t - t'| \leq |t| + |t'| \lt f(q) \leq \varepsilon_{p_0}, \] so \((t - t', q)\) lies in the product neighborhood \((-\varepsilon_{p_0}, \varepsilon_{p_0}) \times W_{p_0}\) on which the slice-chart conclusion \(\Phi(s, q) \in S \iff s = 0\) holds. From \(\Phi(t - t', q) \in S\), this forces \(t - t' = 0\), that is, \(t = t'\). Substituting back into the original equation gives \(\Phi(t, q) = \Phi(t, q')\), with both \(q\) and \(q'\) in \(S\) and \(\Phi(0, q) = q\), \(\Phi(0, q') = q'\). At \(t = 0\) the equation reduces to \(q = q'\). For \(t \neq 0\), applying the flow group law in the same way as before reduces the equation \(\Phi(t, q) = \Phi(t, q')\) to \(\Phi(0, q) = \Phi(0, q')\), again giving \(q = q'\). The restriction \(\Phi|_{\mathcal{O}_\delta}\) is therefore injective.

Suppose \(S\) has codimension one in \(M\). The parameter space \(\mathcal{O}\) has dimension \(1 + k = 1 + (n - 1) = n = \dim M\). The map \(\Phi|_{\mathcal{O}_\delta}\) is by (a) a smooth immersion and by (c) injective, and its source and target have the same dimension. A smooth immersion between manifolds of equal dimension is a local diffeomorphism by the local-diffeomorphism characterization, and is in particular an open map. Being injective as well, it is a diffeomorphism onto its image, which is an open submanifold of \(M\).

Suppose first that \(V_p = 0\). The constant curve \(\gamma : \mathbb{R} \to M\) with \(\gamma(t) \equiv p\) has zero velocity at every point, and the constant zero coincides with \(V_{\gamma(t)} = V_p = 0\) at every \(t\). Hence \(\gamma\) is an integral curve of \(V\) starting at \(p\), defined on all of \(\mathbb{R}\). No open interval strictly contains \(\mathbb{R}\), so \(\gamma\) admits no extension and is therefore maximal. The fundamental theorem identifies \(\theta^{(p)}\) as the unique maximal integral curve of \(V\) starting at \(p\), so \(\theta^{(p)} = \gamma\). In particular \(\mathcal{D}^{(p)} = \mathbb{R}\), and the trajectory \(\theta^{(p)}\) is the constant curve at \(p\).

Now suppose \(p\) is regular, and argue the contrapositive of the immersion conclusion. Assume \(\theta^{(p)}\) fails to be an immersion at some \(s \in \mathcal{D}^{(p)}\), so that \((\theta^{(p)})'(s) = 0\). Let \(q = \theta^{(p)}(s) = \theta_s(p)\). The integral-curve equation gives \[ V_q = V_{\theta^{(p)}(s)} = (\theta^{(p)})'(s) = 0 , \] so \(q\) is a singular point of \(V\). By the singular case just proved, \(\mathcal{D}^{(q)} = \mathbb{R}\) and \(\theta^{(q)}(t) \equiv q\). The hypotheses for the group law of the flow are now in place. We have \(s \in \mathcal{D}^{(p)}\) by assumption, and \(t - s \in \mathcal{D}^{(q)} = \mathbb{R}\) automatically. Therefore, for every \(t \in \mathcal{D}^{(p)}\), \[ \theta^{(p)}(t) = \theta_t(p) = \theta_{t - s} \bigl( \theta_s(p) \bigr) = \theta_{t - s}(q) = q . \] In particular \(\theta^{(p)}\) is constant on \(\mathcal{D}^{(p)}\), so \(V_p = (\theta^{(p)})'(0) = 0\). This contradicts the assumption that \(p\) is regular. The trajectory \(\theta^{(p)}\) is therefore an immersion at every point of \(\mathcal{D}^{(p)}\), and is smooth as the composition of the smooth embedding \(\iota_p : \mathcal{D}^{(p)} \to \mathcal{D}\) given by \(\iota_p(t) = (t, p)\) with the smooth flow \(\theta\).

We reduce to a configuration in which the flowout theorem applies, then read off the desired coordinates from the inverse of the resulting diffeomorphism.

If no hypersurface \(S\) is supplied with the regular point, construct one. Choose any smooth chart \((U, (x^i))\) centered at \(p\). Since \(V_p \neq 0\), some component \(V^j(p) \neq 0\). Fix such an index \(j\) and let \[ S = \bigl\{ q \in U : x^j(q) = 0 \bigr\} . \] This is an embedded hypersurface containing \(p\) (the level-set criterion for embedded submanifolds identifies it as such, since the differential of the coordinate function \(x^j\) is everywhere nonzero on \(U\)). The condition \(V^j(p) \neq 0\) is exactly the statement that \(V_p\) is not tangent to \(\{x^j = 0\}\) at \(p\). If a hypersurface \(S\) was supplied, we use it directly.

In either case, the transversality condition is open. In a slice chart for \(S\) it states that some component of \(V\) in a direction transverse to the slice is nonzero at the point, and continuity of the component functions carries that condition from \(p\) to a neighborhood. The field \(V\) is therefore nowhere tangent to \(S\) on some neighborhood of \(p\) in \(S\). Shrink \(S\) to this neighborhood if necessary. The flowout theorem then applies, and the codimension-one case (d) provides a sub-flow domain \(\mathcal{O}_\delta \subseteq \mathbb{R} \times S\) and a diffeomorphism \(\Phi : \mathcal{O}_\delta \to W\) onto an open submanifold \(W \subseteq M\) containing \(S\), with \(\Phi_*(\partial/\partial t) = V\) on \(W\).

Choose a smooth local parametrization \(X : \Omega \to S\) of \(S\), where \(\Omega \subseteq \mathbb{R}^{n - 1}\) is open and \(X\) is a diffeomorphism onto an open subset \(W_0 \subseteq S\) containing \(p\). Write the coordinates on \(\Omega\) as \((s^2, \ldots, s^n)\). After shrinking \(\Omega\) and \(\delta\) if necessary so that \(\bigl((-\varepsilon, \varepsilon) \times \Omega\bigr) \subseteq \mathcal{O}_\delta\) under the embedding \((t, s^2, \ldots, s^n) \mapsto (t, X(s^2, \ldots, s^n))\), define \[ \begin{align*} &\Psi : (-\varepsilon, \varepsilon) \times \Omega \to M , \\\\ &\Psi(t, s^2, \ldots, s^n) = \Phi \bigl( t, X(s^2, \ldots, s^n) \bigr). \end{align*} \] Both \(\Phi\) and the auxiliary map \((t, s) \mapsto (t, X(s))\) are diffeomorphisms onto their images. The claim for \(\Phi\) is codimension-one case (d) of the flowout theorem, and the auxiliary map is the product of the identity with the diffeomorphism \(X\). The composition \(\Psi\) is therefore a diffeomorphism onto an open neighborhood of \(p\) in \(M\).

It remains to compute the pushforward of \(\partial/\partial t\) under \(\Psi\). Fix a point \((t_0, s_0) \in (-\varepsilon, \varepsilon) \times \Omega\) and let \(q_0 = X(s_0)\). The chain rule applied to the composition \(\Psi = \Phi \circ (\mathrm{id} \times X)\) gives \[ d\Psi_{(t_0, s_0)}\!\left( \frac{\partial}{\partial t}\bigg|_{(t_0, s_0)} \right) = d\Phi_{(t_0, q_0)} \!\circ d(\mathrm{id} \times X)_{(t_0, s_0)}\! \left( \frac{\partial}{\partial t}\bigg|_{(t_0, s_0)} \right) . \] The product map \(\mathrm{id} \times X\) acts as the identity on the first factor of \((-\varepsilon, \varepsilon) \times \Omega\), so its differential sends \(\partial/\partial t|_{(t_0, s_0)}\) to \(\partial/\partial t|_{(t_0, q_0)}\). The time direction is preserved verbatim, and the spatial directions in \(\Omega\) are sent to spatial directions in \(S\) by \(dX\). The above chain-rule expression therefore reduces to \[ d\Psi_{(t_0, s_0)}\!\left( \frac{\partial}{\partial t}\bigg|_{(t_0, s_0)} \right) = d\Phi_{(t_0, q_0)}\!\left( \frac{\partial}{\partial t}\bigg|_{(t_0, q_0)} \right) . \] The flowout-theorem identity \(d\Phi_{(t_0, q_0)}(\partial/\partial t) = V_{\Phi(t_0, q_0)}\) then yields \(d\Psi_{(t_0, s_0)}(\partial/\partial t) = V_{\Psi(t_0, s_0)}\), so on the image of \(\Psi\) we have the pushforward identity \(\Psi_*(\partial/\partial t) = V\). The inverse \(\Psi^{-1}\) is therefore a smooth chart. Renaming the time coordinate \(t\) to \(s^1\), it provides coordinates \((s^1, \ldots, s^n)\) whose first coordinate vector is \(V = \partial/\partial s^1\). The set \(S\) corresponds in these coordinates to \(\{s^1 = 0\}\), so \(s^1\) is a local defining function for \(S\).