# SOLVED: Killing Vectors on Unit Sphere

1. Jan 8, 2014

### Gavins

\bar{}1. The problem statement, all variables and given/known data
Hi, I want to show that
$\frac{\partial}{\partial \phi}$
is a Killing vector on the unit sphere with metric
$ds^2 = d\theta^2 + \sin^2 \theta d \phi^2$

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

3. 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.

Last edited: Jan 8, 2014