negru
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Hi, I'm trying to understand isometries, for example between S^2 (two sphere) and SO(3).
For this I need to show that the killing vectors for S^2
ds^2={d\theta}^2+sin^2 {\theta} {d\phi}^2.
are:
R=\frac{d}{d\phi}}
S=cos {\phi} \frac{d}{d\theta}}-cot{\theta} sin {\phi} \frac{d}{d\phi}}
T=-sin {\phi} \frac{d}{d\theta}}-cot{\theta} cos {\phi} \frac{d}{d\phi}}
I'm not sure how to use the Killing equation, basically because I am confused by R=\frac{d}{d\phi}} 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 cot{\theta} and sin{\theta}cos{\theta} and others zero, but next what are \frac{dR_a}{dx^b}? And {\Gamma}^k_a_b{R_k} for that matter.
Is \frac{dR_1}{dx^2} just equal to \frac{d^2}{d\phi^2} ?
For this I need to show that the killing vectors for S^2
ds^2={d\theta}^2+sin^2 {\theta} {d\phi}^2.
are:
R=\frac{d}{d\phi}}
S=cos {\phi} \frac{d}{d\theta}}-cot{\theta} sin {\phi} \frac{d}{d\phi}}
T=-sin {\phi} \frac{d}{d\theta}}-cot{\theta} cos {\phi} \frac{d}{d\phi}}
I'm not sure how to use the Killing equation, basically because I am confused by R=\frac{d}{d\phi}} 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 cot{\theta} and sin{\theta}cos{\theta} and others zero, but next what are \frac{dR_a}{dx^b}? And {\Gamma}^k_a_b{R_k} for that matter.
Is \frac{dR_1}{dx^2} just equal to \frac{d^2}{d\phi^2} ?