# $\pi\circ \exp|_\mathfrak{m}$ surjective on noncompact homogeneous space

Let $$(G,\langle\,,\,\rangle)$$ a connected Lie group with bi-invariant semi-Riemannian metric. Let $$H$$ be a closed subgroup, and denote by $$h$$ the left-invariant metric on the homogeneous space $$N=G/H$$ such that $$\pi:(G,\langle\,,\,\rangle)\to (N,h)$$ is a semi-Riemannian submersion.
Under these conditions we have an orthogonal decomposition of the Lie-algebra $$\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$$, where $$\mathfrak{h} = Lie(H)$$, and in particular $$N$$ is reductive. Further, the geodesics of $$(N,h)$$ through $$o=eH$$ are exactly $$\gamma(t) = \pi \circ exp(tx)$$, with $$x\in \mathfrak{m}$$.

Is it now true that for $$G$$ noncompact $$\pi\circ \exp|_\mathfrak{m}$$ is surjective, i.e. $$G/H = \pi(\exp\mathfrak{m})$$? How could one prove it?

In the case where $$G$$ is compact it is easy, because then $$N=G/H$$ is compact and the geodesics through $$o$$ hit every other point on the manifold.