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?

**Added** I have since learnt that for the special case of $M$ a symmetric space, then injectivity is indeed equivalent to bijectivity.  This follows from Theorem 1.1 in [this paper of Yannick Voglaire][3]'s.  That helps, but I am interested in the reductive (non-symmetric) case as well.


  [1]: http://www.numdam.org/item?id=BSMF_1957__85__113_0
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=97462
  [3]: https://arxiv.org/abs/1303.5925