Harmonic 3-Forms on Compact Homogeneous Spaces

Abstract

The third real de Rham cohomology of compact homogeneous spaces is studied. Given M= G/ K with G compact semisimple, we first show that each bi-invariant symmetric bilinear form Q on g such that Q| k×k= 0 naturally defines a G-invariant closed 3-form HQ on M, which plays the role of the so called Cartan 3-form Q([· , ·] , ·) on the compact Lie group G. Indeed, every class in H3(G/ K) has a unique representative HQ. Second, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., b3(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 HQ to be g-harmonic, in terms of algebraic invariants of G/K. As an application, we obtain that any 3-form HQ is harmonic with respect to the standard metric, although for any other normal metric, there is only one HQ up to scaling which is harmonic. Furthermore, among a suitable (2 s- 1) -parameter family of G-invariant metrics, we prove that the same behavior occurs if k is abelian: either every HQ is g-harmonic (this family of metrics depends on s parameters) or there is a unique g-harmonic 3-form HQ (up to scaling). In the case when k is not abelian, the special metrics for which every HQ is g-harmonic depend on 3 parameters.