Harmonic morphisms from homogeneous spaces of positive curvature (Q2927887)

Earlier, the authors have shown that any Riemannian symmetric space \((G/H, g)\), which is neither \(G_2/\mathrm{SO}(4)\) nor its non-compact dual, carries local complex-valued harmonic morphism. Even global solutions exist if the symmetric space is of non-compact type [the authors, ibid. 147, No. 2, 389--408 (2009; Zbl 1181.53054)]. In this paper they extend the investigation to more general Riemannian homogeneous spaces. Based on the former result, they give the following one. NEWLINENEWLINENEWLINETheorem 3.3. Let \(\pi:(G/K, \hat{g})\to (G/H,g)\) be a homogeneous Riemannian fibration. If the base \(G/H\) is a symmetric space \((G/H, g)\), but neither \(G_2/\mathrm{SO}(4)\) nor its non-compact dual, then the homogeneous space \(G/K\) admits a local complex-valued harmonic morphism. If \(G/H\) is of non-compact type then global solutions exist.NEWLINENEWLINENEWLINEThe main result of this paper is the following.NEWLINENEWLINENEWLINETheorem 1.1. Let \((M,g)\) be a Riemannian homogeneous space of positive curvature, which is not the Berger space \(\mathrm{Sp}(2)/\mathrm{SU}(2)\).Then \(M\) admits a local complex-valued harmonic morphism.