Consider a reductive homogeneous space $M=G/H$ with corresponding Lie algebra decomposition $\frak{g}=\frak{m}+\frak{h}$. Then there is a local diffeomorphism
$$ (exp\, X, h)\mapsto (exp\, X) h\quad \text{where} \> X\in {\frak{m}},\> h\in H.\qquad (*) $$
Is there an example of a compact reductive homogeneous space satisfying the following two conditions:
The curvature of the canonical connection is not zero.
The local diffeomorphism given in (*) is global.
I can show that any example will have to be non-symmetric and an $H$-space.
Condition 1 can be replaced with the equivalent condition
1.$^\prime$ $\quad exp\, \frak{m}$ is not a subgroup of $G$.
