Isotropic Manifolds

Here's a proof that all isotropic manifolds are homogeneous. Given any $p$ and $q$ in $M$, let $\gamma:[0,2]\rightarrow M$ be a minimizing geodesic from $p$ to $q$. Set $r = \gamma(1)$. So, following $\gamma'(1)$ along for one unit of time lands you at $q$ while following it backwards for one unit o ftime lands you at $p$.

By assumption, there is an isometry $f$ for which $d_r f$ maps $\gamma'(1) \in T_r M$ to $-\gamma'(1)$. Then, by uniqueness of geodesics, we have \begin{align*} f(q) &= f(\exp_r ( \gamma'(1)))\\ &= \exp_r(d_r f \, \gamma'(1)) \\ &= \exp_r(-\gamma'(1)) \\ &= p.\end{align*}

As other have pointed out, among, say, compact simply connected homogeneous spaces, isotropic spaces are very rare. In fact, given such a homogeneous space $G/H$ (where we assume wlog $H$ and $G$ share no common normal subgroups of positive dimension), this space is isotropic iff the induced action of $H$ on $T_{eH} G/H$ is transitive on the unit sphere. Under these assumptions, one can prove, for example, that the universal cover of $H$ has at most two factors. (In fact, those $H$ which act effectively and transitively on a sphere have been completely classified.)

I don't know of a classification of when $G/H$ has $H$ acting transitively on the unit sphere, but beyond $\mathbb{C}P^n$, it also happens for $\mathbb{H}P^n$ and $\mathbb{O}P^2$ (the compact, rank one, symmetric spaces). I don't know any other examples of homogeneous spaces for which $H$ acts transitively on the sphere.

There is a class of spaces variously known as (1) rank one symmetric spaces and/or (2) two-point homogeneous spaces. These are isotropic (and of course homogeneous). A key example to think about are complex projective spaces and complex hyperbolic spaces. Using some of these terms as keywords you should be able to make more progress. Certainly Helgason's book is a must.