Killing vector fields on S2

[alexriemann]

Homework Statement

I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$.

- Solve the Killing equation for the $(\theta,\theta)$ components, that is $(\mathcal L_\xi g)_{\theta\theta}=0$ and show that it gives a Killing vector field $\xi^{(1)}$ that is only a function of $\phi$, $\xi^{(1)}=F(\phi)$. Show that this function has to be periodic and assume that it has the simple form $\xi^{(1)}=A\sin(\phi-\phi_0)$

- Integrate the other Killing component equations and obtain two more Killing vector fields $\xi^{(2)}$ and $\xi^{(3)}$ that will depend on integration constant $\phi_0, c_1,c_2$. By giving these constants simples values, recover the **Killing operators** $\xi_1,\xi_2,\xi_3$ given in the directions associated to the **Killing vector fields** $\xi^{(1)},\xi^{(2)},\xi^{(3)}$.

Relevant Equations

$$\mathcal L_\xi g=0$$

$$\xi_1=(-\sin\phi,-\cot\theta\cos\phi)$$
$$\xi_2=(\cos\phi,-\cot\theta\sin\phi)$$
$$\xi_3=(0,1)$$

I know how to solve this problem by considering the Killing equation, namely $\mathcal L_\xi g=0$ that gives three differential equations involving the components of $\xi=(\xi^\theta,\xi^\phi)$ that can be integrated. The result I get, which I know to be true because this is a common result that can be found anywhere on the web, is:

$$\xi^\theta=\nu\cos\phi-\mu\sin\phi$$
$$\xi^\phi=\delta-\cot \theta(\mu\cos\phi+\nu\sin\phi)$$

Where $\mu,\nu,\delta$ are integration constants. By setting these constants to $(\mu,\nu,\delta)=\{(1,0,0),(0,1,0),(0,0,1)\}$, I obtain three independent vector fields that constitute a basis for the Killing Lie algebra on $S^2$. These are, in the chart:

$$\xi_1=(-\sin\phi,-\cot\theta\cos\phi)$$
$$\xi_2=(\cos\phi,-\cot\theta\sin\phi)$$
$$\xi_3=(0,1)$$

So, this is no big deal. However, the directions of my assignment insist that we strictly stick to the following procedure to find $\xi_1,\xi_2,\xi_3$ (these are given in the directions and match the vectors I wrote above) which is the source of my confusion:

- Solve the Killing equation for the $(\theta,\theta)$ components, that is $(\mathcal L_\xi g)_{\theta\theta}=0$ and show that it gives a Killing vector field $\xi^{(1)}$ that is only a function of $\phi$, $\xi^{(1)}=F(\phi)$. Show that this function has to be periodic and assume that it has the simple form $\xi^{(1)}=A\sin(\phi-\phi_0)$

- Integrate the other Killing component equations and obtain two more Killing vector fields $\xi^{(2)}$ and $\xi^{(3)}$ that will depend on integration constant $\phi_0, c_1,c_2$. By giving these constants simples values, recover the **Killing operators** $\xi_1,\xi_2,\xi_3$ given in the directions associated to the **Killing vector fields** $\xi^{(1)},\xi^{(2)},\xi^{(3)}$.

I must admit that I am very confused with what my teacher is saying. He is basically saying that solving the three Killing component equations $(\mathcal L_\xi g)_{\theta\theta}=(\mathcal L_\xi g)_{\theta\phi}=(\mathcal L_\xi g)_{\phi\phi}=0$ give three Killing vector fields that depend on integration constants whereas, to my understanding, those three equation are solved for the two components of one generic vector field $\xi=(\xi^\theta,\xi^\phi)$ that give rise to three independent Killing vector fields when the integration constants are given some values.

For example, the first Killing component equation reads : $\partial_\theta\xi^\theta=0$, which tells us that the $\theta$-component of the Killing vector field is a function that only depends on $\phi$. This is very different compared to saying that this equation gives a Killing vector field, isn't it?

Can someone make sense of what my teacher is trying to say or is that just wrong overall? I really do think that the directions are not only confusing but wrong. Any insight would be very much appreciated.