# Squashed spheres in general dimension

The point of this question is to help me find references regarding squashed spheres in general dimension. I am interested in the general theory of squashing for arbitrary dimension. All of the references or applications to squashing that I've seen thus far refer to specific dimensions, such as 3 or 7, which are rather exceptional from a Mathematical point of view. I am not sure if the formalism developed in these dimensions can be straightforwardly applied to any dimension, or if the situation is more complicated.

This is purely a mathematical subject, but it has applications in gravity and string theory.

Note: for those who don't know, squashing is a technical term, and not a heuristic one. Squashing is a special type of deformation that preserves topology and the coset structure of the manifold. For example in 3 dimensions the sphere has SO(4) isometry group, but squashing reduces it to SU(2).

• What exactly would you want to learn about squashed spheres? – Danu Jan 3 '16 at 16:38
• I want to know if spheres can be squashed in any dimension, and how they may be squashed. – Surgical Commander Jan 3 '16 at 16:42
• That's still not very clear to me: Do you mean to ask if ellipsoids exist in arbitrary dimension? Isn't that kind of clear? Or do you mean something else by "squashed sphere"? – Danu Jan 3 '16 at 16:52
• Squashing here is a technical term. Ellipsoids are not squashed spheres, for example, squashed spheres only exist in dimensions $\gt 3$. Squashing is a special type of deformation that preserves topology and the coset structure. For example in 3 dimensions the sphere has SO(4) isometry group, but squashing reduces it to SU(2) $\simeq$ SO(3). – Surgical Commander Jan 3 '16 at 16:59
• Ah, thanks. I guess it'd be nice if you could find a source to link to, so others don't have to go through the confusion about the meaning of "squashed spheres" (the link you have right now is not very good, because the first few results are about ellipsoids). – Danu Jan 3 '16 at 17:10