Jacobi equations on minimal homogeneous submanifolds in homogeneous Riemannian spaces
type: Item
instance of: scholarly article
title: Jacobi equations on minimal homogeneous submanifolds in homogeneous Riemannian spaces
zbMATH Open document ID: 0714.58010
DOI: 10.1007/BF01077705
author: Le Khong Van
publication date: 1990-01-01T00:00:00Z
review text: For a minimal submanifold N in a Riemannian manifold M the second variation of the volume of N is given by an integral involving the Jacobi operator I acting on the sections in the normal bundle of N. The index of I is the sum of the multiplicities of the negative eigenvalues of I. The author treats the case where \(M=G/K\) is a Riemannian homogeneous space and \(N=H/K\cap H\). The Jacobi operator is calculated in algebraic terms. Now one can apply Frobenius reciprocity to decompose the space of sections in the normal bundle into finite dimensional I-invariant subspaces. These techniques are used to classify all compact subgroups of the classical compact Lie groups G that are stable minimal submanifolds. Here G carries the biinvariant metric. Finally, the space \(M=SU_{m+1}/T_ m\) is studied where \(T_ m\subset SU_{m+1}\) is a maximal torus. For certain N lower estimates for the index of I are derived.
reviewed by: Christian Bär
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 4176040
zbMATH Keywords: minimal submanifold
zbMATH Keywords: second variation
zbMATH Keywords: volume
zbMATH Keywords: Jacobi operator
zbMATH Keywords: Riemannian homogeneous space
MaRDI profile type: Publication
Wikidata QID: Q115394311
cites work: Complexity of action of reductive groups
cites work: Q4040208
cites work: Q3796286
cites work: Approximate method of solution of queuing problems
cites work: Q3040034
cites work: Q4301559
cites work: Q4734435
cites work: Stability of Minimal Orbits
cites work: On stable currents and their application to global problems in real and complex geometry
cites work: Q3772875
cites work: Minimal varieties in Riemannian manifolds
cites work: The Second Variation Formula for Harmonic Mappings
cites work: Degree Theory on Oriented Infinite Dimensional Varieties and the Morse Number of Minimal Surfaces Spanning a Curve in R n . Part I: n ≥4
cites work: The geometry and structure of isotropy irreducible homogeneous spaces
cites work: On normal homogeneous Einstein manifolds
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4176040