let $X,Y:M\to TM$ be vector fields on $M$. $\nabla\_XY$ is the change in $Y$ along the flow curves of $X$. so for a point $p \in M$ let $\phi^X(t):\mathbb{R}\to M$ be a flow curve of $X$ passing through $p$ :
$$(\phi^X)'=X \; \; ; \; \; \phi^X(0)=p$$
since I can't measure the change in $Y$ at two different points directly. I'll define a parallel transport map $\Pi\_{tX}:T\_pM\to T\_{\phi^X(t)}M$ taking a vector at $p$ to its equivalent at $\phi^X(t)$. Then the derivative at $p$ is:
$$\nabla\_XY|\_p=\lim\_{t\to 0}\frac{\delta\_XY|\_{\phi^X(t)}}{t}=\lim\_{t\to 0}\frac{Y\_{\phi^X(t)}-\Pi\_{tX}(Y\_p)}{t}$$
I tried to do the usual approximation trick by assuming that for a "small" enough $t$ :
$$\nabla\_XY|\_p \approx \frac{\delta\_XY|\_{\phi^X(t)}}{t}$$
but there is a problem that LHS and RHS are not at the same point.


here is what I did "correct me if i'm wrong":


since the points $p$ and $\phi^X(t)$ are so close:
$$\Pi\_{tX}(\nabla\_XY|\_p) \approx \frac{\delta\_XY|\_{\phi^X(t)}}{t}$$
in other words:
$$\Pi\_{tX}(\nabla\_XY|\_p)=\frac{\delta\_XY|\_{\phi^X(t)}}{t}+\vec\epsilon\_1$$
$\vec\epsilon$ being a really small difference vector. Another thing that follows is that the values of the vector field $\nabla\_XY$ at the two points are close so:
$$\Pi\_{tX}(\nabla\_XY|\_p)=\nabla\_XY|\_{\phi^X(t)}+\vec\epsilon\_2$$
then
$$\delta\_XY|\_{\phi^X(t)}=t\nabla\_XY|\_{\phi^X(t)}+t\vec\epsilon\_2-t\vec\epsilon\_1$$
since $\vec\epsilon$ errors are already small. $t\vec\epsilon$ would be even much smaller "second order error" and could be neglected:
$$\delta\_XY|\_{\phi^X(t)} \approx t\nabla\_XY|\_{\phi^X(t)}$$
so:
$$\Pi\_{tX}(Y\_p) \approx (-t\nabla\_XY+Y)|\_{\phi^X(t)}$$


so is there anything wrong with this attempt?



Approximating the parallel transport map on a curve with the covariant derivative
Defining
\begin{equation}
\nabla\_XY|\_p := \lim\_{t\to 0} \frac{\Pi\_{tX}^{-1}(Y\_{\phi^X(t)})-Y\_p}{t}
\end{equation}
the approximation
$\nabla\_XY|\_p \approx \frac{\Pi\_{tX}^{-1}(Y\_{\phi^X(t)})-Y\_p}{t}$, for $t$ small, holds and now both sides are tangent vectors in $T\_pM$. It also follows that $\Pi\_{tX}^{-1}(Y\_{\phi^X(t)})\approx t\,\nabla\_XY|\_p+Y\_p$, namely $Y\_{\phi^X(t)}\approx \Pi\_{tX}(t\,\nabla\_XY|\_p+Y\_p)$ and therefore
\begin{equation}
\Pi\_{tX}(Y\_p)\approx Y\_{\phi^X(t)}-t\,\Pi\_{tX}(\nabla\_XY|\_p).
\end{equation}


