Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives.
Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ such that $V_p$ is open in $T_pM$, w.r.t. to the scalarproduct $<.,.>_p$.
Maybe I did too much math today, that I can't find a way to prove it, but is it now possible to find a subset $H \subset G$, such that $H.V_p$ is open in $TM$?