SOLVED: Killing Vectors on Unit Sphere

[Gavins]

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Homework Statement

Hi, I want to show that
$\frac{\partial}{\partial \phi}$
is a Killing vector on the unit sphere with metric
$ds^2 = d\theta^2 + \sin^2 \theta d \phi^2$

Homework Equations

I compute the Christoffel symbols to be
$\Gamma^\theta_{\phi \phi} = -\sin \theta \cos \theta$
$\Gamma^\phi_{\phi \theta} = \cot \theta$

The Attempt at a Solution

Then computing Killing's equation for the theta-phi component,
$\nabla_\phi X_\theta + \nabla_\theta X_\phi$

This gives
$X_{\theta,\phi} + X_{\phi, \theta} - 2 \cot \theta X_\phi$

But this doesn't give 0 since $X_\phi \neq 0$. Where did I go wrong?

EDIT:
Nvm. Forgot about the lowered index.

 

[lippyka]

Computing the Killing equation for the lowered index component, \nabla_\phi X^\theta + \nabla_\theta X^\phi This gives X^{\theta}_{,\phi} + X^{\phi}_{, \theta} + 2 \cot \theta X^\phi Which is 0 since all the terms are 0.