Homogeneous Sphere Bundles and the Isotropic Riemann Manifolds

Let M be a connected metric space and H an isometry group of M which leaves fixed a point p in M. M is said H isotropic at p when, for any two points q and r of M at the same distance from p, there exists an isometry in H which carries q to r. When H coincides with the maximum isometry group leaving p fixed, M is said merely isotropic at p.