Spectrally distinguishing symmetric spaces II

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.