Levi-Civita connection and parallel transport

Question

Suppose $(M,g)$ is a Riemannian manifold, $∇$ is a linear connection. We can define the parallel transport $\left\{T^{\gamma}\right\}$ as a family of linear isomorphisms, I know that the connection $∇$ is compatible with $g$ if and only if all of its parallel transport $T^{\gamma}_{t_0,t}$ are isometries between $T_{\gamma(t_0)}M$ and $T_{\gamma(t)}M$. My question is if the connection $∇$ is Levi-Civita connection? If the connection is torsion-free, the parallel transport must satisfies a stronger condition. Like if we only consider the holononmy group $Hol_{p}(M,∇)$, use the condition that $∇$ compatible with the metric, we know that it is a subgroup of $O(T_{p}M)$. So what can the property of torsion-free tell us ? In order to define the parallel transport, the key point is to solve a system of linear ODEs. I think the torsion-free property can give a restriction to this system of linear ODEs. In other words, the system of linear ODEs can be simplified. But it seems difficult to be reflected on the level of linear isomorphisms between tangent space, since we only use the existence and uniqueness of solutions.