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.
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