Harmonic morphisms from homogeneous spaces of positive curvature
type: Item
instance of: scholarly article
title: Harmonic morphisms from homogeneous spaces of positive curvature
zbMATH Open document ID: 1307.53050
DOI: 10.1017/S0305004114000334
author: Sigmundur Gudmundsson
author: Martin Svensson
published in: Mathematical Proceedings of the Cambridge Philosophical Society
publication date: 2014-11-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1312.6515
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 6365834
zbMATH Keywords: harmonic morphism
zbMATH Keywords: homogeneous space
zbMATH Keywords: symmetric space
MaRDI profile type: Publication
arXiv ID: 1312.6515
cites work: Harmonic morphisms between Riemannian manifolds
cites work: Harmonic morphisms from complex projective spaces
cites work: The normal homogeneous space (𝖲𝖴(3)×𝖲𝖮(3))/𝖴^{∙}(2) has positive sectional curvature
cites work: Q3945267
cites work: Compact homogeneous Riemannian manifolds with strictly positive curvature
cites work: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
cites work: Homogeneous Riemannian spaces of four dimensions
cites work: Harmonic morphisms from solvable Lie groups
cites work: Die Krümmung des Raumes \(Sp(2)/SU(2)\) von Berger
OpenAlex ID: W3098112021
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review text: 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.
reviewed by: Weihuan Chen
description: scientific article; zbMATH DE number 6365834