Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity

Four-dimensional field equations are derived for perturbations of the seven-sphere size and squashing parameter in eleven-dimensional supergravity configurations in which the rank-four tensor gauge field strength is proportional to the four-dimensional volume element. Field equations are also derived for perturbations in the size of a round or squashed Einstein-metric $S^7$ and in the magnitude of gauge-field-potential components proportional to an antisymmetric torsion tensor on these seven-spheres. The known anti—de Sitter—invariant solutions with round or squashed seven-spheres are shown to be classically stable to these perturbations.