Let's choose a point and a tangent vector $x\in U\subseteq M$, $\omega(x)\in T_{x}M$ in a local chart of a manifold. If we define a curve $\gamma : [0, 1]\rightarrow U$ such that $$\left.\frac{d\gamma}{dt}\right|_{t = 0} = \left.\frac{d\gamma}{dt}\right|_{t = 1} = \omega(x),$$ then when we pull-back the parallel-transport, we won't be getting the same tangent vector, of the intristic curvature of the manifold. This is known as general holonomy of Riemannian manifolds.
My question is, how differently does holonomy behave along geodesics and arbitrary curves?
