Sur les déformations et la stabilité des solutions de la Supergravité en 11 dimensions

The authors consider the spontaneously compactified solutions of Freund and Rubin of 11-dimensional supergravity and study their stability with respect to the fluctuations of the metric and the auxiliary field $A$. The basic solution is an 11-dimensional Riemannian manifold on which is defined a 3-form, called a "Maxwell" auxiliary field $A$. This solution, which is a direct product of two homogeneous spaces, is isometrically imbedded in a 13-dimensional Euclidean space; the sphere $S^7$ is imbedded in $E^8$ and the anti-de Sitter space in $E^5$. Small perturbations are assumed not to change the global topology of the imbedded manifold and they are then conceived of as imbedding deformations. At each point of the imbedded manifold these variations can be decomposed into tangential deformations, which are nothing else than diffeomorphisms, and into normal deformations which really change the manifold. It is proved that the normal fluctuations respecting the global topology of the compactified solution are composed uniquely of conformal factors of the anti-de Sitter space and of the solutions of the Klein-Gordon equation with real mass, which at finite distance have an exponential decay to zero