Sigmundur Gudmundsson

On the geometry of tangent bundles

Abstract The main aim of this survey paper is to write a detailed and unified presentation of some of the best known results on the geometry of tangent bundles of Riemannian manifolds.

On the existence of harmonic morphisms from symmetric spaces of rank one

SummaryIn this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍHn, ℂHn, ℝH2t+1, ℍPn, ℂPn and ℝP2n+1 of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases.

Harmonic morphisms between spaces of constant curvature

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let n: U-*N be a horizontally homothetic harmonic morphism. In this paper we show that if n has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometries of M and N one of six well known examples.

Harmonic morphisms from complex projective spaces

In this paper we study harmonic morphisms ∅ :U ⊂ ℂPm →N2 from open subsets of complex projective spaces to Riemann surfaces. We construct many new examples of such maps which are not holomorphic with respect to the standard Kähler structure on ℂPm.

On the geometry of harmonic morphisms

Let π:M→B be a horizontally conformal submersion. We give necessary curvature conditions on the manifolds M and B, which lead to non-existence results for certain horizontally conformal maps, and harmonic morphisms. We then classify all such maps between open subsets of Euclidean spaces, which additionally have totally geodesic fibres and are horizontally homothetic. They are orthogonal projections on each connected component, followed by a homothety.

Minimal submanifolds of hyperbolic spaces via harmonic morphisms

In this paper we give a method for constructing complete minimal submanifolds of the hyperbolic spaces Hm. They are regular fibres of harmonic morphisms from Hm with values in Riemann surfaces.

Non-holomorphic harmonic morphisms from Kähler manifolds

We give a method for constructing non-holomorphic harmonic morphisms from Kähler manifolds. The method is then used to obtain many such examples defined locally on the complex Grassmannians.

Harmonic morphisms from solvable Lie groups

In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

Harmonic morphisms from quaternionic projective spaces

In this paper we give a method for constructing harmonic morphisms from quaternionic projective spaces ℍPk with values in a Riemann surface.

Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.