Rescaling of the metric, invariant operators, and spontaneous compactification

A criterion for enlargement of the symmetry of a $G$-invariant metric on $G/H$ to the symmetry of a $G'$-invariant metric on $G'/H'$ is formulated and proved. The complete set of vacuum solutions of $d=11$ supergravity on $S^7$ and the spectrum of the Laplace–Beltrami operator for each of the solutions are obtained. The degeneracy of the spectrum of the invariant operators resulting from rescaling with symmetry enlargement is investigated.