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Spectral isolation of bi-invariant metrics on compact Lie groups

Carolyn S. Gordon; Dorothee Schueth; Craig J. Sutton — 2010

Annales de l’institut Fourier

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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