I'm currently working in a particular conformal compactification/completion of the Minkowski space-time, but I'm stuck at showing that the exponential of every vector field in it is globally defined. I've worked in the general case, since it's essentially the same as the particular case with $p=3$ and $q=1$.
I have a $p+q$-dimensional manifold, $M$, on which the orthogonal group $O(p+1,q+1)$ acts transitively. So we know that for every $m\in M$,
$$
M\cong O(p+1,q+1)/O(p+1,q+1)_m,
$$ where $O(p+1,q+1)_m$ is the stabilizer of $m$. My question is:
- How can I show/see that the exponential of every vector field in $M$ is globally defined?
Edit: I also know that $M\cong \mathbb{S}^p\times\mathbb{S}^q /\mathbb{Z}_2$, if it helps.
The fact that you're looking at a homogeneous space is irrelevant -- what's relevant is that $M$ is compact (since you said it's homeomorphic to $\mathbb S^p\times \mathbb S^q/\mathbb Z_2$). On a compact manifold, every vector field is complete, meaning its flow exists for all time.