This is part of another old qualifying exam problem, so any help is appreciated! We are given two vector fields $X$ and $Y$ on $\mathbb{R}^3$ described by
$$X = \frac{\partial}{\partial y} + z\frac{\partial}{\partial x}\qquad Y = \frac{\partial}{\partial z} + y\frac{\partial}{\partial x}$$
and we are asked to give a parameterization $(x(s,t), y(s,t), z(s,t))$ for the (unique) surface passing therough the point $p = (1,0,0)$ and tangent to the vector fields $X$ and $Y$ at each point, and to then give an equation of the form $F(x,y,z) = 0$ for this surface.
I was able to determine that $[X, Y] = 0$, and that $X$ and $Y$ were linearly independent at $p$, so I computed the flows $\theta_t$ of $X$ and $\phi_s$ of $Y$ to be
$$\begin{align*}
\theta_t(x,y,z) &= (xe^{zt}, y + t, z)\\
\phi_s(x,y,z) &= (xe^{ys}, y, z + s)
\end{align*}$$
which gives us
$$\theta_t\circ \phi_s = \phi_s\circ \theta_t = (xe^{zt + ys + st}, y + t, z + s),$$ but it is at this point that I get stuck. To solve this, I have tried to use the following theorem from Lee's textbook:
Theroem 9.46: Let $M$ be a smooth $n$ manifold, and let $(V_1, \ldots, V_k)$ be a linearly independent $k$-tuple of smooth commuting vector fields on an open subset $W\subseteq M$. For each $p\in W$, there exists a smooth coordinate chart $(U, (s^i))$ centered at $p$ such that $V_i = \frac{\partial}{\partial s^i}$ for $i = 1,\ldots,k$. If $S\subseteq W$ is an embedded codimension-$k$ submanifold and $p$ is a point of $s$ such that $T_pS$ is complementary to the span of $(V_1\vert_p,\ldots,V_k\vert_p)$, then the coordinates can also be chosen such that $S\cap U$ is the slice defined by $s^1=\cdots=s^k=0$.
At $p = (1,0,0)$ we have that $\text{span}\left\{X\vert_p,Y\vert_p\right\} = \text{span}\left\{\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right\}$, so if we take $S$ to be the $x$-axis, then we can apply the theorem. However, since $S$ is not just a point, I am having trouble finding the explicit parameterization $(x(s,t), y(s,t), z(s,t))$. Agian, any help on how I need to go about this is greatly appreciated. Thank you!
Best Answer
One can do this without directly thinking about or computing flows.
Hint Taking a general $1$-form $$\omega_0 := p(x, y, z) \,dx + q(x, y, z) \,dy + r(x, y, z) \,dz$$ on $\Bbb R^3$, imposing $\omega(X) = \omega(Y) = 0$, and solving for the coefficient functions $p, q, r$, shows that the annihilator of $X$ and $Y$ at each point is spanned by $$\omega_0 := dx - z \,dy - y \,dz,$$ but this form is exact: $$\omega_0 = d(x - y z) .$$ (Involutivity of the distribution $D := \operatorname{span}\{X, Y\}$ implies that any annihilator $\omega$ of $D$ satisfies $d\omega = \theta \wedge \omega$ for some $1$-form $\theta$, and since $\Bbb R^3$ is simply connected, if $\omega$ vanishes nowhere there is a function $f$ such that $f \omega$ is exact.)