Harmonic morphisms from four-dimensional Lie groups

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.

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Harmonic morphisms from the classical non-compact semisimple Lie groups

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2008.10.003 ◽

2009 ◽

Vol 27 (1) ◽

pp. 47-63 ◽

Cited By ~ 1

Author(s):

Sigmundur Gudmundsson ◽

Anna Sakovich

Keyword(s):

Lie Groups ◽

Harmonic Morphisms ◽

Semisimple Lie Groups

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Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2005.12.003 ◽

2006 ◽

Vol 24 (4) ◽

pp. 351-366 ◽

Cited By ~ 1

Author(s):

Sigmundur Gudmundsson ◽

Martin Svensson

Keyword(s):

Lie Groups ◽

Harmonic Morphisms ◽

Semisimple Lie Groups

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Harmonic morphisms from solvable Lie groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109002564 ◽

2009 ◽

Vol 147 (2) ◽

pp. 389-408 ◽

Cited By ~ 4

Author(s):

SIGMUNDUR GUDMUNDSSON ◽

MARTIN SVENSSON

Keyword(s):

Lie Groups ◽

Symmetric Spaces ◽

Global Solutions ◽

Riemannian Symmetric Space ◽

Continuous Family ◽

Harmonic Morphisms ◽

Solvable Lie Groups ◽

3 Dimensional ◽

Riemannian Symmetric Spaces ◽

Simply Connected

AbstractIn 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.

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