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dg.differential geometry – ambit of divergence operator on the house of traceless symmetric $(0,2)$ tensors; conformal vector fields on an capricious metric on $S^2$ retort

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dg.differential geometry – ambit of divergence operator on the house of traceless symmetric $(0,2)$ tensors; conformal vector fields on an capricious metric on $S^2$

Let $gamma$ breathe a metric on $S^2$.
I’m attempting to decipher the next PDE on a $(0,2)$ symmetric traceless tensor $A$:
$$div_{gamma} A = omega$$
the place $omega$ is a 1-figure.

It is thought that there exists a unique resolution if and provided that $omega$ is $L^2$ orthogonal to conformal killing (CK) vector fields on $(S^2,gamma)$ (the equivocate algebra of the conformal group of $S^2$).

It is just too recognized that if $gamma$ is the spherical sphere, then the house of CK vector fields is a 6-dim house.

If $gamma$ just isn’t the spherical metric, what can we are saying concerning the house of CK vector fields? Will it too breathe 6-dimensional? I arbitrator that if $gamma$ is of traverse curvature, then it will not admit any CK vector fields and so the divergence operator is an isomorphism onto the house of 1-forms. What about if $gamma$ is of optimistic curvature?

Using the uniformization theorem, we all know there exists an $f$ (in a three-dimensional house I arbitrator?) such that $gamma = f^2 gamma_0$ the place $gamma_0$ is the spherical sphere, and so $div_{gamma} = frac{1}{f^2} div_{gamma_0}$. So we necessity $f^2 omega$ to breathe $L^2$ orthogonal to conformal killing vector fields on the spherical sphere. Can we impoverish the freedom of selecting $f$ to challenge out exactly which $omega$ will labor?

It appears to me that for any CK vector bailiwick $X$ with respect to $gamma_0$, we maintain $mathcal{L}_X gamma = f^2 mathcal{L}_X gamma_0 + X(f^2) gamma_0 = (h + frac{X(f^2)}{f^2})gamma$ the place $h$ satisfies $mathcal{L}_X gamma_0 = h gamma_0$. So $X$ is CK with respect to $gamma$ too. But that appears to say that the house of CK vector fields would not cipher on the metric. That should breathe incorrect as a result of $gamma$ vitality breathe of traverse curvature, which does not admit CK vector fields. What am I doing incorrect? (There might be one thing basic that I’m not judgement).

Any help or references is appreciated.

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