Recall that a Riemannian manifold $(M,g)$ is isotropic if for any $p\in M$ and any unit vectors $v,w\in T_pM$ there is an isometry $f:M\to M$ such that $f_\ast(v)=w.$ Recall also that $(M,g)$ is homogeneous if for any $p,q\in m$ there is an isometry $f:M\to M$ such that $f(p)=q.$
I would appreciate answers to any of the following questions:
I know that the simply connected constant curvature spaces are isotropic. What are some other examples of isotropic manifolds?
I suspect that the flat torus is not isotropic and that the product metric on $M\times N$ is not isotropic if $M$ and $N$ are not isometric. Are these true?
Is there some way to relate transitivity of the holonomy group on the unit sphere in $T_pM$ to isotropy?
Are isotropic manifolds and symmetric spaces related?
What are some results connecting homogeneity and isotropy? Homogeneous isotropic manifolds are important in relativity; are these classified, at least in dimension 3? When can we conclude that a homogeneous manifold is isotropic? When can we conclude that an isotropic manifold is homogeneous?
What are some references for reading about isotropic manifolds? I've tried using google and a few Riemannian geometry and general relativity books (Petersen, Wald, Choquet-Bruhat) but I haven't been able to find very much.