Harmonic 3-Forms on Compact Homogeneous Spaces

The third real de Rham cohomology of compact homogeneous spaces is studied. Given M = G/K with G compact semisimple, we first showthat each bi-invariant symmetric bilinear form Q on g such that Q vertical bar k x k = 0 naturally defines a G-invariant closed 3-form H-Q on M, which plays the role of the so called Cartan 3-form Q([center dot, center dot], center dot) on the compact Lie group G. Indeed, every class in H-3(G/K) has a unique representative H-Q. Second, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., b(3)(G/K) = s - 1 if G has s simple factors, we give the conditions to be fulfilled by Q and a given G-invariant metric g in order for H-Q to be g-harmonic, in terms of algebraic invariants of G/K. As an application, we obtain that any 3-form H-Q is harmonic with respect to the standard metric, although for any other normal metric, there is only one H-Q up to scaling which is harmonic. Furthermore, among a suitable (2s - 1)-parameter family of G-invariant metrics, we prove that the same behavior occurs if k is abelian: either every H-Q is g-harmonic (this family of metrics depends on s parameters) or there is a unique g-harmonic 3-form H-Q (up to scaling). In the case when k is not abelian, the special metrics for which every H-Q is g-harmonic depend on 3 parameters.

Automated summary

This article studies the third real de Rham cohomology of compact homogeneous spaces. It is shown that for a compact semisimple group G, a bi-invariant symmetric bilinear form Q on the Lie algebra g can naturally define a G-invariant closed 3-form H-Q on the homogeneous space G/K, which plays a role similar to the Cartan 3-form on G. The conditions for Q and a given G-invariant metric g to make H-Q g-harmonic are given in terms of algebraic invariants of G/K. The behavior of g-harmonic 3-forms H-Q under different metrics is also explored.