How does the Torsion of two vector fields act on their corresponding flows?

Question

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?

Answer 1

Back in Élie Cartan's papers (but I don't remember specifically which) there is a discussion of the torsion as the translational part of the holonomy of an affine connection, whereas curvature is the rotational part (in the Riemannian case). You should also read this discussion on mathoverflow.

Answer 2

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 $Y$.