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On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative along a left-invariant vector field of an harmonic form is again a harmonic form. This affirmation is on the context compact Lie group case with bi-invariant Riemmanian metric. I have some questions:

How can I prove this statement? Do you have a reference? Should I try first to prove that the codifferential(the adjoint of the exterior derivative) commutes with the Lie derivative?

Does it hold in general? That is, if I have a complete Riemannian manifold, does it hold that the Lie derivative of a harmonic form is again a harmonic form?

Can you suggest some references?

Thank you very much.

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  • $\begingroup$ This is a pointwise computation that, with enough struggling, you might be able to do yourself. The Hodge Laplacian is defined using only the exterior derivative and Hodge star operators. You already know that exterior differentiation commutes with Lie differentiation. So it remains to figure out what happens when you commute the star operator with the Lie derivative. I suggest working this out for 0-forms and 1-forms first. Don't try anything fancy at first. Just grind it out using local coordinates, unless you see how to do it better another way. $\endgroup$
    – Deane Yang
    Commented Sep 4, 2016 at 20:06
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    $\begingroup$ Not True in general. Consider for example the harmonic function $f(x,y)=x^2-y^2$ on $\mathbb{R}^2$. The Lie derivative of $f$ along the vector field $x\partial_x$ is $2x^2$. This function is not harmonic. Helgason must have included a an additional condition on the vector field. Very likely it is a condition that implies that teh flow of the vector field preserves the metric. In this case the computation is trivial. $\endgroup$
    – Liviu Nicolaescu
    Commented Sep 4, 2016 at 20:08
  • $\begingroup$ The Lie derivative is along a left-invariant vector field. I just eddited the question with this information. But I still don't understand how this can be helpfull. $\endgroup$
    – Max Reinhold Jahnke
    Commented Sep 4, 2016 at 20:20
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    $\begingroup$ On a Lie group the flow of a left invariant vector field preserves right-invariant metrics. On a compact Lie group any right-invariant metric is bi-invariant. Thus the flow of your vector field preserves the metric, any any flow that preserves the metric maps harmonic forms to harmonic forms. Helgason's claim should be obvious in view of the definition of Lie derivative in terms of flows. $\endgroup$
    – Liviu Nicolaescu
    Commented Sep 4, 2016 at 21:08
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    $\begingroup$ @LiviuNicolaescu on a compact Lie group, it's not true that every right-invariant metric is bi-invariant. On a connected compact Lie group, given a scalar product $b$ on the Lie algebra, you get a unique right-invariant metric by right translation. This is also left-invariant iff $b$ is conjugation invariant. This is a nontrivial condition as soon as the group is not abelian. $\endgroup$
    – YCor
    Commented Sep 4, 2016 at 21:33

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