Computing the Lie brackets of vector fields on a specific Riemannian manifold

Question

Let $\mathcal{H}^3=\lbrace (x,y,z) \in \mathbb{R}^3 \mid z>0\rbrace$ be equipped with the Riemannian metric: \begin{equation*} g=\frac{dx^2+dy^2+dz^2}{z^2} \end{equation*} And consider the vector fields: \begin{equation*} A=z\frac{\partial}{\partial x}, \quad B=z\frac{\partial}{\partial y}, \quad C=z\frac{\partial}{\partial z} \end{equation*} I want to compute the Lie brackets $[A,B]$,$[B,C]$ and $[A,C]$ and express them as constant linear combinations of $A$,$B$ and $C$, but I can't figure out how to compute them? Should I use a test function $f$ and a point $p \in \mathcal{H}^3$ or can I do it without?

Answer 1

First, the metric does not matter for this kind of problem, only the underlying differential structure is used.

Second, you could very well use a test function and apply the result at an arbitrary point, but it is slightly shorter without:

$$[B,C] = [z \partial_y, z \partial_y] = z \partial_y (z \partial_z) - z \partial_z (z \partial_y) = z^2 \partial_y \partial_z - z \cdot 1 \cdot \partial_y - z^2 \partial_z \partial_y = -z \partial_y = -B$$

(I have used Leibniz's rule when applying partial derivative operators and I have also used the symmetry of second order partial derivatives).