Let $G$ be a finite-dimensional connected real Lie group and $\mathfrak{g}$ its Lie algebra. Let $\exp : \mathfrak{g} \to G$ denote the exponential map. Among the results proved independently by [Dixmier][1] and [Saito][2] more than half a century ago is that if $\exp$ is injective, it is also surjective, which happens only if $G$ is a solvable Lie group. My question is not about Lie groups but about their (reductive) homogeneous spaces. So let $G$ be as above (i.e., finite-dimensional, real, connected) and let $H \subset G$ be a closed subgroup with Lie algebra $\mathfrak{h}$. For simplicity of exposition let us assume that $\mathfrak{h}$ has a complementary subspace $\mathfrak{m}$ in $\mathfrak{g}$ which is stable under the adjoint action of $H$. Let $M = G/H$ be the corresponding homogeneous space with $\pi: G \to M$ the natural surjection. The composition $\sigma = \pi \circ \exp : \mathfrak{m} \to M$ is a local diffeomorphism from a neighbourhood of $0 \in \mathfrak{m}$ to a neighbourhood of the identity coset in $M$. Indeed, a choice of basis for $\mathfrak{m}$ gives *exponential coordinates* in a neighbourhood of the identity in $M$. I am interested in the case that $\sigma$ is everywhere injective. **Question** When is $\sigma$ injective? In particular, can I conclude that in that case it is also surjective? [1]: http://www.numdam.org/item?id=BSMF_1957__85__113_0 [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=97462