A question about tangent subbundle

Question

Let $x$ be a manifold, $E$ is a subbundle of $TX$ , my question is :

Can you give example such that vector fields $\xi ,\eta$ lie in $E$,but bracket $[\xi ,\eta]$ does not lie in $E$ in some point of $x$.

Answer 1

Consider the tangent space to $R^3$, which we can write as $R^3 \times R^3$. At each point $(x, y, z)$, consider the plane orthogonal to $(-y, x, 1)$. These planes form a subbundle.

Now look at two vector fields, one radial from the origin, $$ F(x, y, z) = \begin{bmatrix}x\\ y\\ 0\end{bmatrix} $$ the other "circumferential", i.e., $$ G(x, y) = \begin{bmatrix}-y\\ x\\ -(x^2+y^2)\end{bmatrix}. $$

(I had originally written $$ G(x, y, z) = [-y, x, 0]. $$ which doesn't even lie in the specified plane, and the OP complained that the bracket, for that $G$, turns out to be zero. D'oh!)

As the OP noted in comments, in coordinates, the bracket is just \begin{align} [F, G] &= G' F - F' G\\ &= \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -2x & -2y & 0 \end{bmatrix} \begin{bmatrix} x\\ y\\ 0\end{bmatrix} -\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} -y\\ x\\ -(x^2+y^2)\end{bmatrix} \\ &= \begin{bmatrix} -y\\ x\\ -2(x^2 + y^2) \end{bmatrix} - \begin{bmatrix} -y\\ x\\ 0 \end{bmatrix}\\ &= \begin{bmatrix} 0\\ 0\\ -2(x^2 + y^2) \end{bmatrix} \end{align}

That bracket is nonzero, and points in the $z$-direction, i.e., a direction not in the plane (except at the origin, of course).