Construction des applications harmoniques non rigides d'un tore dans la sphère
type: Item
instance of: scholarly article
title: Construction des applications harmoniques non rigides d'un tore dans la sphère
zbMATH Open document ID: 0532.53039
DOI: 10.1007/BF02330008
author: Gábor Tóth
published in: Annals of Global Analysis and Geometry
publication date: 1983-01-01T00:00:00Z
review text: A harmonic map f: \(M\to S^ n\) of a compact Riemannian manifold M into the Euclidean n-sphere \(S^ n\) is said to be infinitesimally rigid if every projectable divergence-free Jacobi field v along f is of the form \(v=X{\mathbb{O}}f\), where \(X\in so(n+1)\) is an infinitesimal isometry on \(S^ n\). Here we show that the vector function \^v: \(M\to {\mathbb{R}}^{n+1}\) induced by such v satisfies the equation \(\Delta^ M \hat v=2 e(f) \hat v\), where e(f) is the energy density of f. Using this we prove that every harmonic embedding of \(S^ 1\times S^ 1\) into \(S^ n\) with \(e(f)=1/2\) is infinitesimally non-rigid.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 3844568
zbMATH Keywords: harmonic map
zbMATH Keywords: Jacobi field
zbMATH Keywords: rigidity
zbMATH Keywords: energy density
zbMATH Keywords: harmonic embedding
MaRDI profile type: Publication
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: A Report on Harmonic Maps
cites work: On Homotopic Harmonic Maps
cites work: Q5658761
cites work: Q5731354
cites work: Q2550561
cites work: Rigidity of certain harmonic mappings
full work available at URL: https://doi.org/10.1007/bf02330008
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description: scientific article; zbMATH DE number 3844568