I'm having a little trouble with this problem for Lee - Introduction to Smooth Manifolds (2nd ed). The problem is as follows (Problem 19-5):
Let $D$ be the distribution of $\mathbb{R}^3$ spanned by \begin{align*} X&=\frac{\partial}{\partial x}+yz\frac{\partial}{\partial z}, & Y&=\frac{\partial}{\partial y}. \end{align*}
- Find an integral submanifold of $D$ passing through the origin.
- Is $D$ involutive? Explain your answer in light of the first part.
After thinking about the problem for a moment, I've determined that the surface $z=0$ (the $xy$-plane) fits the bill for the first question. But this seems too easy, so I'm naturally skeptical. Would it be beneficial to come up with a nontrivial example (if one exists)?
I don't believe that $D$ is involutive, since $$ [X,Y]=X(1)\frac{\partial}{\partial y}-Y(1)\frac{\partial}{\partial x}-Y(yz)\frac{\partial}{\partial z}=-z\frac{\partial}{\partial z}, $$ and (I'm quite sure) $-z\,(\partial/\partial z)$ cannot be written as a linear combination of $X$ and $Y$, so $[X,Y]$ is not a local section of $D$, and hence it is not involutive.
I guess "the explanation" that the 2nd part of the last question is asking for is just the acknowledgement that just because a distribution isn't integrable, doesn't mean there doesn't exist some point which is still contained in an integral submanifold?
Am I missing something here? I usually expect the exercises in the text to be more involved than this, so I'm just skeptical.
Any clarification would be much appreciated. Thank you.
