Soit un groupe de Lie compact et connexe, et soit une métrique bi-invariante sur . On démontre que est isolée spectralement dans la classe des métriques invariantes à gauche : plus précisément, il existe un entier positif tel que, dans un voisinage de dans la classe des métriques invariantes à gauche et de volume inférieur ou égal à celui de , la métrique est determinée de manière unique par les premières valeurs propres strictement positives de son Laplacien (sans multiplicités). Si est simple, on peut choisir .
We show that a bi-invariant metric on a compact connected Lie group is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric on there is a positive integer such that, within a neighborhood of in the class of left-invariant metrics of at most the same volume, is uniquely determined by the first distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where is simple, can be chosen to be two.
@article{AIF_2010__60_5_1617_0, author = {Gordon, Carolyn S. and Schueth, Dorothee and Sutton, Craig J.}, title = {Spectral isolation of bi-invariant metrics on compact Lie groups}, journal = {Annales de l'Institut Fourier}, volume = {60}, year = {2010}, pages = {1617-1628}, doi = {10.5802/aif.2567}, zbl = {1203.53035}, mrnumber = {2766225}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2010__60_5_1617_0} }
Gordon, Carolyn S.; Schueth, Dorothee; Sutton, Craig J. Spectral isolation of bi-invariant metrics on compact Lie groups. Annales de l'Institut Fourier, Tome 60 (2010) pp. 1617-1628. doi : 10.5802/aif.2567. http://gdmltest.u-ga.fr/item/AIF_2010__60_5_1617_0/
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