Isometric actions and harmonic morphisms (Q5954543)

Harmonic morphisms are maps between Riemannian manifolds which preserve (local) harmonic functions. They were characterised as horizontally weakly conformal (a geometrical condition generalising Riemannian submersions) harmonic maps. They are submersive almost everywhere and their fibres form a conformal foliation of the domain space. The purpose of the author is to study and classify harmonic morphisms by looking at the foliations which, at least locally, are the fibres of a harmonic morphism. NEWLINENEWLINENEWLINEIf the codimension of the foliation is 2, then a foliation produces harmonic morphisms if and only if it is conformal and with minimal leaves. When the codimension is not 2, \textit{R. L. Bryant} [Commun. Anal. Geom. 8, 219-265 (2000; Zbl 0966.53045)] showed that a one-dimensional Riemannian foliation will produce harmonic morphisms if and only if it is locally generated by Killing fields. The aim of this paper is to consider Riemannian foliations locally generated by Killing fields of dimension greater than one. NEWLINENEWLINENEWLINEThe first step is to show that a Riemannian foliation locally generated by Killing vector fields produces harmonic morphisms if and only if \(\roman{trace}(\roman{ad} I)=0\), where \(I\) is the integrability tensor of the horizontal distribution. When the foliation is generated by a closed subgroup of \(\roman{Isom}(M,g)\), \(\roman{trace}(\roman{ad} I)\) takes a clearer expression. The rest of the paper consists of a series of cases where a Riemannian foliation is locally generated by Killing vector fields and satisfies \(\roman{trace}(\roman{ad} I)=0\), notably, if the horizontal distribution is integrable, if \(\roman{trace}\circ\roman{ad} =0\) (with a partial converse), if the leaves are naturally reductive or locally symmetric. Besides, for the case where the foliation is generated by a closed subgroup of \(\roman{Isom}(M,g)\), a list of situations for which \(\roman{trace}(\roman{ad} \mathfrak{g}) =0\) (this implies \(\roman{trace}\circ\roman{ad} =0\)) is given: nilpotent or Abelian, semi-simple, compact, general linear. NEWLINENEWLINENEWLINEThe article closes with some examples, in particular between the general linear group and the Grassmannian and Stiefel manifolds.