Hi, I'm trying to understand isometries, for example between S^2 (two sphere) and SO(3).(adsbygoogle = window.adsbygoogle || []).push({});

For this I need to show that the killing vectors for S^2

[tex]ds^2={d\theta}^2+sin^2 {\theta} {d\phi}^2.[/tex]

are:

[tex]R=\frac{d}{d\phi}}[/tex]

[tex]S=cos {\phi} \frac{d}{d\theta}}-cot{\theta} sin {\phi} \frac{d}{d\phi}} [/tex]

[tex]T=-sin {\phi} \frac{d}{d\theta}}-cot{\theta} cos {\phi} \frac{d}{d\phi}} [/tex]

I'm not sure how to use the Killing equation, basically because I am confused by [tex]R=\frac{d}{d\phi}}[/tex] not being a vector? How do I calculate the comma derivative of R then? I suppose I could convert to cartesian coordinates or something, but there has to be a direct way.

I can get that some components of the Christoffel symbol are [tex]cot{\theta}[/tex] and [tex]sin{\theta}cos{\theta}[/tex] and others zero, but next what are [tex]\frac{dR_a}{dx^b}[/tex]? And [tex]{\Gamma}^k_a_b{R_k}[/tex] for that matter.

Is [tex]\frac{dR_1}{dx^2}[/tex] just equal to [tex]\frac{d^2}{d\phi^2}[/tex] ?

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# Killing vector on S^2

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