Geometric interpretation of splitting of sequence associated to a homogeneous space Let $G$ be a Lie group acting transitively on a smooth manifold $M$.  Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\xi : \mathfrak{g} \to \Gamma(TM)$ be the Lie algebra homomorphism sending $X \in \mathfrak{g}$  to the generating vector field $\xi_X$.   Fix a point $p \in M$ and consider the evaluation $\operatorname{ev}_p : \mathfrak{g} \to T_p M$ sending $X \in \mathfrak{g}$ to $\xi_X(p)$.  The kernel of this map is the Lie algebra $\mathfrak{h}_p$ of the stabiliser $H_p$ of $p$ in $G$.
This gives rise to a short exact sequence
$$
0 \longrightarrow \mathfrak{h}_p \longrightarrow  \mathfrak{g}  \longrightarrow  T_pM  \longrightarrow  0
$$
I'm particularly interested in the case when the homogeneous space is not reductive, so that the sequence does not split as $\mathfrak{h}_p$-modules.
My question is the following:
What is the geometrical interpretation of a (vector space) splitting $T_p M \to \mathfrak{g}$ of the above exact sequence?
I suspect that this is standard, but I cannot find much literature in the non-reductive case.
 A: I am not aware of a general "geometric interpretation" of such a vector space splitting in the non-equivariant case, but there is at least one interesting application, which is along the lines of the comment by @FriedrichKnop. Namely, you get a family of distinguished "normal" coordinates and a family of distinguished curves in $M$. Identifying $M=G/H$, let $\mathfrak m\subset\mathfrak g$ be the image of the splitting, i.e. a linear subspace complementary to $\mathfrak h$. Denoting by $\pi:G\to G/H$ the natural projection you see that $X\mapsto \pi(g\cdot exp(tX))$ restricts to a local diffeomorphism from an open neighbourhood of $0$ in $\mathfrak m$ onto an open neighbourhood of $\pi(g)$ in $G/H$. In each point, this gives you a family of local coordinates parametrized by the fiber over the point. The projections of the exponential curves make sense for all times, so you get a family of distinguished parametrized curves. This is a family of reasonable size, because fixing the initial velocity, there is just a finite dimensional freedom. 
To see that this is interesting in examples, consider the cases of $S^n$ as a homogeneous space of $SO(n+1,1)$ or of $S^{2n+1}$ as a homogeneous space of $SU(n+1,1)$. In both cases, you get a natural vector space splitting for $\mathfrak g$, which is not $H$-equivariant. In the first case, the resulting curves are the conformal circles on $S^n$. In the second case, you can get the chains for the natural CR structure on $S^{2n+1}$ as distinguished curves of that type (with a slight additional normalization).  
