Let $M$ be a Riemannian manifold and $G$ a closed connected subgroup of isometries of $M$. Call the pair $(M,G)$ an isotropic pair if $G$ acts transitively on the sphere bundle $SM$. As an example, the pair $(S^6,SO(7))$ is isotropic, but also $(S^6,G_2)$.
I am looking for a reference for the classification of all isotropic pairs. In several places in the literature, the possible manifolds $M$ are given (in the non-flat case these are symmetric spaces of rank one), but not the corresponding groups $G$.The best reference seems to be J. Tits paper Espaces homogènes et isotropes, et espaces doublement homogènes from 1954, but it is not so easy to understand his terminology.
Questions:
- Is the following list complete?
- There seems to be no negatively curved versions of the pairs $(S^6,G_2)$ and $(S^7,Spin(7))$. What is the reason for this?
Flat case:
- $\mathbb{R}^n$ with the group generated by translations and $SO(n), U(n/2), SU(n/2), Sp(n/4), Sp(n/4) U(1), Sp(n/4) Sp(1)$
- $\mathbb{R}^7$ with the group generated by translations and $G_2$
- $\mathbb{R}^8$ with the group generated by translations and $Spin(7)$
- $\mathbb{R}^{16}$ with the group generated by translations and $Spin(9)$
Symmetric space case (each with $G$ equal to the connected component of the trivial element of the isometry group):
- $S^n$ sphere
- $\mathbb{RP}^n$ real projective space
- $\mathbb{H}^n$ real hyperbolic space
- $\mathbb{CP}^n$ complex projective space
- $\mathbb{CH}^n$ complex hyperbolic space
- $\mathbb{HP}^n$ quaternionic projective space
- $\mathbb{HH}^n$ quaternionic hyperbolic space
- $\mathbb{OP}^2$ octonionic projective plane
- $\mathbb{OH}^2$ octonionic hyperbolic plane
Other cases:
- $(\mathbb{RP}^6,G_2)$ and $(S^6,G_2)$
- $(\mathbb{RP}^7,Spin(7))$ and $(S^7,Spin(7))$