Narrow Results

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

We construct four-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not Kähler. We then prove that the Riemannian Lie groups constructed are not Einstein manifolds. This answers an important open question in the theory of complex-valued harmonic morphisms from Riemannian 4-manifolds.

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called pseudo harmonic morphisms, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Finally, we construct a canonical f-structure associated to every map satisfying (PHWC) and find conditions on this f-structure to ensure the harmonicity of the map.

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

We study slant submersions and conformal slant submersions from nearly Kaehler manifolds onto Riemannian manifolds and investigate conditions for such maps to be totally geodesic maps. We also obtain conditions for a slant submersion and a conformal slant submersion from a nearly Kaehler manifold onto a Riemannian manifold to be a harmonic map and a harmonic morphism, respectively.

In this paper, we study biharmonic morphisms–the maps between Riemannian manifolds which pull back germs of biharmonic functions to germs of biharmonic functions. We obtain characterizations of such maps and some results concerning their links to harmonic morphisms and biharmonic maps.