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.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.