Distribution

Distribution and Integral Flow

Previously, in §Vector Fields, we saw that for a given manifold \(M\) and any \(C^\infty\) vector field \(X\) defined on it, there exists a sufficiently small \(\epsilon>0\) such that the following equation

\[\sigma'(t)=X(\sigma(t)),\qquad \sigma(0)=p\tag{1}\]

is satisfied by a curve \(\sigma:(-\epsilon,\epsilon)\rightarrow M\). If \(X_p\neq 0\), then for the curve \(\sigma\) defined in this way, its image in \(M\), \(S\), can be viewed as a submanifold of \(M\) containing the point \(p\).

Meanwhile, equation (1) above enforces not only the image of \(\sigma\) but also the way it is parametrized. In contrast, since the submanifold \(S\) is determined independently of the parametrization of \(\sigma\), it is determined not by the vector \(X_p\), but solely by the 1-dimensional subspace of \(T_pM\), \(\span(X_p)\).

Definition 1 Suppose an \(m\)-dimensional manifold \(M\) is given, and, satisfying \(1\leq k\leq m\), an integer \(k\) is given. A function that assigns to each \(p\in M\) a subspace of \(T_pM\), namely a \(k\)-dimensional subspace \(\mathcal{D}(p)\), via \(p\mapsto\mathcal{D}(p)\), is called a \(k\)-dimensional distribution.

A \(k\)-dimensional distribution \(\mathcal{D}\) is \(C^\infty\) if for each \(p\in M\), there exist a suitable open neighborhood \(U\) and \(C^\infty\) vector fields \(X_1,\ldots, X_k\) defined on it such that for each \(x\in U\), the following equation

\[\mathcal{D}(x)=\span\{(X_1)_x,\ldots, (X_k)_x\}\]

holds.

A nowhere-vanishing vector field \(X\), as examined above, defines a 1-dimensional distribution through the map \(p\mapsto\span(X_p)\subseteq T_pM\). Then the submanifold \(S\) is determined so as to satisfy the following equation

\[T_xS=\mathcal{D}(x)\qquad\text{for all $x$}\]

Therefore, we define as follows.

Definition 2 On \(M\), for a \(k\)-dimensional distribution \(\mathcal{D}\) defined on it, a manifold satisfying the following equation

\[T_xS=\mathcal{D}(x)\qquad\text{for all $x$}\]

namely \(S\), is called an integral manifold of \(\mathcal{D}\).

If for each \(p\in M\), there exists an integral manifold containing \(p\), it is called an integrable distribution.

Frobenius Theorem

The following theorem is well known.

Theorem 3 (Frobenius) On a manifold \(M\), consider a \(k\)-dimensional \(C^\infty\) distribution \(\mathcal{D}\). Then \(\mathcal{D}\) is integrable if and only if, for any \(X,Y\in\mathcal{D}\), \([X,Y]\in\mathcal{D}\) holds.

Moreover, for any \(k\)-dimensional involutive distribution, the following holds:

1. For each \(p\in M\), there exists, containing \(p\), an integral manifold of \(\mathcal{D}\).

2. Furthermore, by suitably choosing a coordinate system centered at \(p\), \((U,\varphi)\), the slices defined by the following equations

   \[x^i=\text{constant},\qquad i>k\]

   can be made to be integral manifolds of \(\mathcal{D}\).

3. Finally, if \(\Phi:N\rightarrow M\) is a connected integral manifold and \(\Phi(N)\subseteq U\), then \(\Phi(N)\) is contained in only one of the slices from (2).

A distribution satisfying the latter condition is called involutive. Therefore, the Frobenius theorem can be stated in short as saying that a distribution \(\mathcal{D}\) is integrable if and only if it is involutive.

One direction of this theorem can be proved quite easily.

Lemma 4 On a manifold \(M\), suppose that a smooth distribution \(\mathcal{D}\) is integrable. Then \(\mathcal{D}\) is an involutive distribution.

Proof

Let \(X,Y\in\mathcal{D}\) and pick a point \(p\in M\). We must show that \([X,Y]_p\in\mathcal{D}(p)\).

Since \(\mathcal{D}\) is an integrable distribution, containing \(p\), there exists an integral submanifold of \(\mathcal{D}\), \(\Phi:S\rightarrow M\). Let \(s\in S\) be a point satisfying \(\Phi(s)=p\). For any \(x\in S\), since

\[\dd{\Phi_x}:T_xS\rightarrow\mathcal{D}(\Phi(x))\]

is an isomorphism, we can find, satisfying for all \(x\in S\)

\[\dd{\Phi_x}(\tilde{X}_x)=X_{\Phi(x)},\qquad \dd{\Phi_x}(\tilde{Y}_x)=Y_{\Phi(x)}\]

two vector fields \(\tilde{X},\tilde{Y}\). Then, since these and \(X,Y\) are \(\Phi\)-related vector fields respectively, by §Lie Derivative, ⁋Proposition 9, \([\tilde{X},\tilde{Y}]\) and \([X,Y]\) are \(\Phi\)-related. Therefore

\[[X,Y]_p=\dd{\Phi_s}([\tilde{X},\tilde{Y}]_s)\in\mathcal{D}(p)\]

holds.

Thus, the difficult part of the proof of Theorem 3 (Frobenius) is the converse direction. This proceeds by induction on the dimension \(k\) of the distribution.

Lemma 5 Let an \(m\)-dimensional manifold \(M\), a point \(p\in M\), and, satisfying \(X_p\neq 0\), a vector field \(X\) be given. Then, containing \(p\), there exists a suitable coordinate system \((U,\varphi), \varphi=(x^1,\ldots, x^m)\) such that

\[X\vert_U=\frac{\partial}{\partial x^1}\bigg\vert_U\]

holds.

Proof

Centered at the point \(p\), choose a coordinate system \((V,\tau), \tau=(y^1,\ldots, y^m)\) such that

\[X_p=\frac{\partial}{\partial y^1}\bigg\vert_p\]

holds. Without loss of generality, we may assume that \(V\) is sufficiently small so that for a suitable \(\epsilon>0\), the function

\[(-\epsilon,\epsilon)\times V\rightarrow M;\qquad(t,q)\mapsto \phi^t(q)\]

is well-defined and \(C^\infty\). (§Vector Fields, ⁋Theorem 6) Moreover, if we choose \(\epsilon>0\) and \(W\) small enough that the inclusion

\[\phi^t(\tau^{-1}(\{0\}\times W))\subseteq V\quad(\lvert t\rvert<\epsilon),\qquad \text{$W$ is an open neighborhood of the origin in $\mathbb{R}^{m-1}$}\]

holds, then the function

\[\sigma: (-\epsilon,\epsilon)\times W\rightarrow M;\qquad (t,a^2,\ldots, a^m)\mapsto \phi^t(\tau^{-1}(0,a^2,\ldots, a^m))\]

is well-defined. But since

\[\dd{\sigma}\left(\frac{\partial}{\partial r^1}\bigg\vert_0\right)=\frac{\partial}{\partial y^1}\bigg\vert_p=X_p\neq 0,\qquad \dd{\sigma}\left(\frac{\partial}{\partial r^i}\bigg\vert_0\right)=\frac{\partial}{\partial y^i}\bigg\vert_p\]

holds, \(\sigma\) is nonsingular at the origin, and thus \(\sigma^{-1}\) defines a coordinate map.

Proof of Theorem 3

Assume that the theorem holds for all \(k-1\)-dimensional distributions, and let \(\mathcal{D}\) be a \(k\)-dimensional distribution. For a point \(p\in M\), we may assume that \(\mathcal{D}\) is, in a neighborhood of \(p\), spanned by \(k\) vector fields \(X_1,\ldots, X_k\). Now, applying Lemma 5, so that

\[X_1\vert_V=\frac{\partial}{\partial y^1}\]

holds, we can find, centered at \(p\), a coordinate system \((V,\tau),\tau=(y^1,\ldots, y^m)\).

Now let us define \(k\) vector fields \(Y_1,\ldots, Y_k\) by the equations

\[Y_1=X_1,\qquad Y_i=X_i-(X_i(y^1))X_1\quad(i\geq 2)\]

. Since the \(X_i\) are mutually independent, so are the \(Y_i\).

Now let \(S\) be the slice defined by \(y^1=0\). Then, by restricting \(Y_2,\ldots, Y_k\) to \(S\), we obtain the vector fields

\[Z_i=Y_i\vert_S \qquad (i\geq 2)\]

. At this time, since the equation

\[Z_i(y^1)=Y_i(y^1)=0\]

holds, we see that the \(Z_i\) are independent vector fields contained in the tangent space of \(S\). Therefore, on \(S\), they span a \(k-1\)-dimensional distribution.

Now, in order to use the induction hypothesis, let us show that this distribution is involutive. That is, we must show that for any \(i,j\), \([Z_i,Z_j]\in\span(Z_2,\ldots, Z_k)\) holds.

Consider the inclusion \(\iota:S\rightarrow M\). Then since the \(Z_i\) and \(Y_i\) are \(\iota\)-related, it suffices to show that \([Y_i,Y_j]\in\span(Y_2,\ldots, Y_k)\). However,

\[Y_i(y^1)=X_i(y^1)-X_i(y^1)X_1(y^1)=X_i(y^1)-X_i(y^1)=0\]

holds for all \(i\geq 2\), and therefore \([Y_i,Y_j]y^1=0\). On the other hand, since \(\mathcal{D}\) is involutive, we can write \([Y_i,Y_j]=\sum_{l=1}^kc_lY_l\), and applying \(y^1\) to this, from \(Y_1(y^1)=1\) we obtain \(c_1=0\). From this we see that the \([Y_i,Y_j]\) actually belong to \(\span(Y_2,\ldots, Y_k)\).

Now, applying the second assertion of the theorem to the involutive distribution on \(S\), \(\span(Z_2,\ldots, Z_k)\), we can choose, centered at \(p\in S\), a coordinate system \((w^2,\ldots, w^m)\) such that the slices obtained from the equations

\[w^i=\text{constant},\qquad i>k\]

become integral submanifolds of \(\span(Z_2,\ldots, Z_k)\).

To complete the proofs of the first and second assertions, let us define \(m\) functions

\[x^1=y^1,\quad x^j=w^j\circ\pi\]

where \(\pi:V\rightarrow S\) is the projection that eliminates the \(y^1\) component. Now, since the \((x^i)\) are independent functions, we know that there exists a coordinate system \((U,\varphi)\) having them as component functions. Then the coordinate system defined this way satisfies the second assertion. That is, the slices defined by the equations

\[x^i=\text{constant},\qquad i>k\]

become integral manifolds of \(\mathcal{D}\). To show this, it suffices to show that for each \(x^{k+1},\ldots, x^m\), the \(Y_i(x^{k+j})\) are all \(0\).

First, from the definition of the \(x^i\), we know that \(\partial x^j/\partial y^1=\delta_{j1}\) holds, and therefore on \(U\),

\[Y_1=\frac{\partial}{\partial x^1}\]

holds. For the remaining \(Y_2,\ldots, Y_k\), first using the equation

\[\frac{\partial}{\partial x^1}Y_i(x^{k+j})=Y_1(Y_i(x^{k+j}))=[Y_1,Y_i]x^{k+j}\]

, from the condition that \(\mathcal{D}\) is involutive, if we apply

\[[Y_1,Y_i]=\sum_{l=1}^k c_{il}Y_l\]

to the right-hand side, we see that

\[\frac{\partial}{\partial x^1}(Y_i(x^{k+j}))=\sum_{l=2}^k c_{il}Y_l(x^{k+j})\]

holds. Now, for a 1-dimensional slice \(W\) obtained by fixing all \(x^2,\ldots,x^m\) to constants, the \(Y_i(x^{k+j})\) are single-variable functions of \(x^1\), and therefore the above equations become \(k-1\) linear homogeneous ODEs, so the solution is uniquely determined by the initial values.

These slices \(W\) meet \(S\cap U\) at only a single point, and since

\[Y_i(x^{k+j})=Z_i(w^{k+j})=0\]

holds here, the proofs of the first and second assertions are complete.

Finally, we must show the third assertion. This time, let \(\pi\) be the projection from \(\mathbb{R}^m\) onto the last \(m-k\) coordinates. Then since the image of \(\mathcal{D}\) under \(\dd{(\pi\circ\varphi)}\) is \(0\),

\[\dd{(\pi\circ\varphi\circ\Phi)}\equiv 0\]

holds for any \(y\in N\). But since \(N\) is connected, \(\pi\circ\varphi\circ\Phi\) is a constant function, and therefore \(\Phi(N)\) is contained in a single slice.

---

References

[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012

---