Spectrally distinguishing symmetric spaces II

TL;DR

This paper proves that for certain symmetric spaces, the spectrum of the Laplace-Beltrami operator uniquely determines the metric up to isometry, establishing spectral rigidity among known homogeneous metrics.

Contribution

It demonstrates spectral uniqueness of symmetric metrics on specific symmetric spaces, showing that the spectrum characterizes the metric up to isometry.

Findings

Spectral rigidity of symmetric metrics on certain spaces.

Spectral uniqueness among known homogeneous metrics.

Non-flat irreducible symmetric spaces are spectrally distinguished.

Abstract

The action of the subgroup $\operatorname{G}_2$ of $\operatorname{SO}(7)$ (resp.\ $\operatorname{Spin}(7)$ of $\operatorname{SO}(8)$) on the Grassmannian space $M=\frac{\operatorname{SO}(7)}{\operatorname{SO}(5)\times\operatorname{SO}(2)}$ (resp.\ $M=\frac{\operatorname{SO}(8)}{\operatorname{SO}(5)\times\operatorname{SO}(3)}$) is still transitive. We prove that the spectrum (i.e.\ the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric $g_0$ on $M$ coincides with the spectrum of a $\operatorname{G}_2$-invariant (resp.\ $\operatorname{Spin}(7)$-invariant) metric $g$ on $M$ only if $g_0$ and $g$ are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.