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 and Saito 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's. That helps, but I am interested in the reductive (non-symmetric) case as well.