On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative of an harmonic form is again a harmonic form. This affirmation is on the 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.
Part II (page 40) of This book of Kentaro Yano, Salomon Bochner would be useful
Curvature and Betti Numbers, K. Yano, S. Bochner.