How does the Torsion of two vector fields act on their corresponding flows? Let $X$ and $Y$ be vector fields defined on an open neighborhhod of a smooth manifold $M$ endowed with an (arbitrary) affine connection $\nabla$ (i'm not assuming anything apart from it being a connection on $TM$). 
I'm trying to understand the torsion $T(X,Y)=\nabla_X Y - \nabla_Y X - [X,Y]$ of the connection in terms of flows. Here's what i have so far:
Denoting the local flows of $X$ and $Y$ by $\varphi^X_t$ and $\varphi^Y_t$ resp. and their commutator by $\alpha(t)= \varphi^Y_{-t} \varphi^X_{-t}\varphi^Y_t\varphi^X_t$. We have the following relation between the lie bracket and the $\alpha$: 
$$[X,Y] = \frac{1}{2} \alpha ''(0)$$
So what i'm left with is finding a way to express $\nabla_X Y - \nabla_Y X$ in terms of flows. Obviously there must be some input from the connection. I tried to compute the flow by exponentiating from the lie algebra of vector fields but i didn't get very far... 
This problem made me realize i have no idea how integral curves and parallel are related. A word about how the they relate to each other would in any case be very helpful. 
Ideally I'd like to have an expression for $\nabla_X Y - \nabla_Y X$ in terms of parallel transport and the flows of $X$ and $Y$. Is there such a charactrization?

So from Samelson Lie Bracket and Curvature, its shown that 
$$ (\nabla_Xs)^v =[X^h,s^v]$$ where $ X \in \Gamma(TM)$ and $s \in \Gamma(E)$ and $\nabla$ is the covariant derivative on $E$. The $s^v$ is the vertical extension and is defined as below 
Like wise the $X^h$ is the horizontal lift of the vector field and is defined as 
So
 $$ \nabla_XY-\nabla_YX = \pi([X^h,Y^v] - [Y^h,X^v]) $$
Where $\pi$ is the vertical projection.Therefore
$$\nabla_XY - \nabla_YX = \pi(\frac12(\beta^{''}(0)-\gamma^{''}(0)))$$
Where $$ \beta = \varphi_{-t}^{Y^v}\varphi_{-t}^{X^h}\varphi_{t}^{Y^v}\varphi_{t}^{X^h}$$ and $$ \gamma = \varphi_{-t}^{X^v}\varphi_{-t}^{Y^h}\varphi_{t}^{X^v}\varphi_{t}^{Y^h}$$
where $\varphi_{t}^{Y^v} $,$ \varphi_{t}^{X^h} $,$ \varphi_{t}^{X^v}$ and $ \varphi_{t}^{Y^h} $ are the flows of $ Y^v $, $X^h$, $X^v$ and $Y^h$ respectively. 
I don't know if this makes any sense but I think by finding a way to horizontally lift and vertically extend the flows you would have a characterization  of the covariant derivative in terms of the flows of $X$ and $Y$.
