Spectral isolation of naturally reductive metrics on simple Lie groups

TL;DR

This paper proves that all naturally reductive metrics on compact simple Lie groups are spectrally isolated and shows that isospectral compact symmetric spaces form a finite set, based on spectral properties.

Contribution

It establishes spectral isolation for all naturally reductive metrics on compact simple Lie groups and demonstrates finiteness of isospectral symmetric spaces.

Findings

All naturally reductive metrics are spectrally isolated.

Any collection of isospectral compact symmetric spaces is finite.

Spectral data can distinguish these geometric structures.

Abstract

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.