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Carolyn S. Gordon; Dorothee Schueth; Craig J. Sutton
Spectral isolation of bi-invariant metrics on compact Lie groups
(Isolation spectrale des métriques bi-invariantes sur les groupes de Lie compacts)
Annales de l'institut Fourier, 60 no. 5 (2010), p. 1617-1628, doi: 10.5802/aif.2567
Article PDF | Reviews MR 2766225 | Zbl 1203.53035
Class. Math.: 53C20, 58J50
Keywords: Laplacian, eigenvalue spectrum, Lie group, left-invariant metric, bi-invariant metric

Résumé - Abstract

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two.

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