Spectrally distinguishing symmetric spaces II

The action of the subgroup G2 of SO(7) (resp. Spin(7) of SO(8)) on the Grassmannian space M = SO(7)/(SO(5)×SO(2)) (resp. M = SO(8)/(SO(5)×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 g0 on M coincides with the spectrum of a G2-invariant (resp. Spin(7)-invariant) metric g on M only if g0 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.