Spectral isolation of bi-invariant metrics on compact Lie groups

TL;DR

This paper proves that on compact Lie groups, bi-invariant metrics are uniquely identified by a finite set of Laplacian eigenvalues among nearby left-invariant metrics, with fewer eigenvalues needed for simple groups.

Contribution

It establishes spectral isolation of bi-invariant metrics within left-invariant metrics on compact Lie groups, including a minimal eigenvalue count for simple groups.

Findings

Bi-invariant metrics are spectrally isolated within left-invariant metrics.

A finite number of eigenvalues suffices to determine the metric locally.

For simple groups, only two eigenvalues are needed.

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.