# 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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