I've been thinking about the following problem:

Equip $\mathbb{R}^3$ with coordinates $(x,y,z)$ and define two vector fields $X$ and $Y$ by

$$

X=\frac{\partial}{\partial x}+f(x,y)\frac{\partial}{\partial z},\hspace{.5 in}\text{and}\hspace{.5 in}Y=\frac{\partial}{\partial y}+g(x,y)\frac{\partial}{\partial z}.

$$

Define the distribution $\Delta\subset T\mathbb{R}^3$ by

$$

\Delta =\text{span}(X,Y).

$$

Determine conditions on the functions $f(x,y)$ and $g(x,y)$ that imply $\Delta$ is involutive. What do your conditions imply about the maximal connected integral submanifolds of $\Delta$?

I've mostly worked out this question, however I'm stuck on the last question: to find how my conditions affect the maximal connected integral submanifolds of $\Delta$.

Here's my solution so far.

**"Solution"**

Note that

\begin{align*}

[X,Y]&=X(1)\frac{\partial}{\partial y}+X(g)\frac{\partial}{\partial z}-Y(1)\frac{\partial}{\partial x}-Y(f)\frac{\partial}{\partial z}\\

&=\left(\frac{\partial g}{\partial x}+f \frac{\partial g}{\partial z}\right)\frac{\partial}{\partial z}-\left(\frac{\partial f}{\partial y}+g\frac{\partial f}{\partial z}\right)\frac{\partial}{\partial z}\\

&=\left(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\right)\frac{\partial}{\partial z},

\end{align*}

where we have used that $f$ and $g$ are independent of $z$. Now $\Delta$ is involutive if and only if $[X,Y]\in\text{span}(X,Y)$, so we require the determinant of

$$

\begin{pmatrix}

1 & 0 & f\\

0 & 1 & g\\

0 & 0 & \frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}

\end{pmatrix}

$$

to vanish identically, i.e. we require

$$

\frac{\partial g}{\partial x}=\frac{\partial f}{\partial y}.

$$

My problem is that any curve following $X$ would be of the form $\alpha(t)=(t+x_0,y_0,F(t))$, where $F$ is some antiderivative of $f(t+x_0,y_0)$, and similiarly for $Y$. Since in this form $f$ is constant in the $y$-direction I don't see how the condition above will come into play. Is my thinking incorrect? Is one of my calculations wrong? Any help would be greatly appreciated. Thanks.

## Best Answer

You found the condition for the commutator of $X$ and $Y$ to be in the span of $X$ and $Y$. This condition does not exhibit itself when you only move along either $X$ or $Y$. But your questions asks about the integral submanifold, not just integral curves of $X$ and $Y$ separately.

When you move first along $X$ for some time $t$ and then along $Y$ for some time $s$, you may not end up at the same position as if you reverse the order. The commutator $[X,Y]$ measures the extent to which the two positions differ when $t$ and $s$ are very small. If the commutator is in the span of $X$ and $Y$, it means that the direction of the offset is in the plane spanned by $X$ and $Y$. Otherwise, you would have left the distribution $\Delta$.

In your example, the commutator vanishes and so there is no offset.