Geometry of coset spaces and massless modes of the squashed seven-sphere in supergravity

We derive some general results for Killing vectors on arbitrary coset manifolds and explicitly exhibit the squashed seven-sphere as the coset space Sp₄×Sp₂/Sp₂×Sp₂. Using these results, we then analyze the zero-mass sector of supergravity of the squashed S⁷ and argue that it is not interpretable as a spontaneously broken version of N=8 supergravity. We also point out the existence of a new solution which combines squashing and torsion.