Deformations of the 4-sphere with 8-dimensional isometry groups

Question

I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.

Answer 1

There cannot be an 8-dimensional group $G$ acting effectively on $S^4$ by Riemannian isometries. The following argument may not be the best, but it explains why this is true. (I will assume that $G$ acts effectively, since, otherwise, we can quotient by the closed subgroup $K\subset G$ that acts trivially, and work with $G/K$, which does act effectively, instead.)

First, such a $G$ cannot act transitively on $S^4$, since the effective transitive Lie group actions on spheres were classified by Borel, and the only effective transitive compact connected group action on $S^4$ is the standard one with $G = \mathrm{SO}(5)$, a 10-dimensional Lie group. [If you don't like relying on Borel's classification, here is a 'low-tech' argument to rule out the transitive case: If a connected compact $d$-dimensional group $G$ acts transitively and effectively on $S^4$, then we can write $S^4 = G/K$, where $K$ is the $G$-stabilizer of a point. This $K$ must be a compact connected subgroup of $\mathrm{SO}(4)$ of dimension $d{-}4$. All the proper connected subgroups of $\mathrm{SO}(4)$ are conjugate to a subgroup of $\mathrm{U}(2)$. So, if $K$ were of dimension less than $6$, then $S^4$ would have a $G$-invariant, almost complex structure. However, $S^4$ does not support an almost complex structure.]

Now look at a generic $G$-orbit $\Sigma$, which will be a smooth submanifold. If it is codimension $1$, then $G$ acts by isometries on $\Sigma$ and the normal subgroup $H\subset G$ that fixes $\Sigma$ pointwise is discrete. However, the dimension of $G/H$ can be at most $6$ (since it acts effectively on a compact Riemannian $3$-manifold), so $G$ has dimension at most $6$. Similarly, if $G$ fixes a point of $S^4$, then $G$ has dimension at most $6$.

If all of the $G$-orbits were of dimension $1$ or of dimension $2$ then this would define a nontrivial splitting of the tangent bundle of $S^4$, which does not exist.

Thus, the $G$-orbits of an $8$-dimensional Lie group acting by isometries on $S^4$ endowed with some Riemannian metric would have dimension either $1$ or $2$ and there would have to be some of each dimension. A 'generic' $G$-orbit $\Sigma$ would be of dimension $2$. If $H\subset G$ is the normal subgroup that fixes $\Sigma$ pointwise, then $H$ has dimension at most $1$ and $G/H$ has dimension at most $3$, so $G$ has dimension at most $4$.