Instability for harmonic foliations on compact homogeneous spaces
type: Item
instance of: scholarly article
title: Instability for harmonic foliations on compact homogeneous spaces
zbMATH Open document ID: 1161.53021
author: Kei Ichikawa
author: Tomonori Noda
published in: Differential Geometry and its Applications
publication date: 2009-02-20T00:00:00Z
review text: Recall that one of the characterizations of a harmonic foliation \(\mathcal F\) of a compact Riemannian manifold is that it is an extremal, or critical point, of the energy functional. In other words, the first variation of energy vanishes for all variations of \(\mathcal F\). A harmonic foliation is said to be unstable if the second variation of energy is negative for some variation of the foliation. In this paper, the authors investigate the instability of harmonic foliations of compact subspaces of Euclidean space and of compact homogeneous spaces. They find sufficient conditions on the geometry of the compact space so that every harmonic foliation is unstable. In the case of submanifolds of Euclidean space, the condition is an inequality between the 2nd fundamental form and the Ricci curvature operator. In the case of homogeneous spaces, it is an inequality between the first eigenvalue of the Laplace operator on functions and the scalar curvature. As a corollary they classify the compact symmetric spaces which admit no non-trivial harmonic foliations.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C43
zbMATH DE Number: 5509729
zbMATH Keywords: harmonic foliation
zbMATH Keywords: stability
zbMATH Keywords: instability
zbMATH Keywords: symmetric space
zbMATH Keywords: homogeneous space
reviewed by: James Hebda
MaRDI profile type: Publication
full work available at URL: j.difgeo.2008.06.012
OpenAlex ID: W2091077119
Wikidata QID: Q115357459
cites work: Stability of foliations with complex leaves on locally conformal Kähler manifolds
cites work: Q3681516
cites work: Infinitesimal automorphisms and second variation of the energy for harmonic foliations
cites work: The Index of Harmonic Foliations on Spheres
cites work: On stable currents and their application to global problems in real and complex geometry
cites work: Q4745012
cites work: Harmonic foliations on a compact Riemannian manifold of non-negative constant curvature
cites work: Stability of harmonic maps and standard minimal immersions
cites work: The Second Variation Formula for Harmonic Mappings
cites work: Minimal immersions of Riemannian manifolds
cites work: Minimal foliations on Lie groups
DOI: 10.1016/J.DIFGEO.2008.06.012
description: scientific article; zbMATH DE number 5509729