Supersymmetry breaking by torsion and the Ricci-flat squashed seven-spheres

In eleven-dimensional supergravity, unbroken supersymmetries may survive spontaneous compactification to V4× V7only for Freund-Rubin solutions to the equations of motion. A torsion which renders V7Ricci-flat and breaks supersymmetry can then be induced spontaneously. Two squashed seven-spheres with torsion are obtained in this way: the flattening torsion components are constant and given by the octonion multiplication table.