2022/10/14 by Lauret, Jorge, Will, Cynthia E. · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2210.07662
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 \mathfrakg such that Q|_\mathfrakk×\mathfrakk=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. Secondly, 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 (2s-1)-parameter family of G-invariant metrics, we prove that the same behavior occurs if \mathfrakk 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 \mathfrakk is not abelian, the special metrics for which every HQ is g-harmonic depend on 3 parameters.