# Parallel Transport Along Nearby Curves Produces Nearby Vectors

Let $(M, g)$ be a Riemannian manifold and $d$ denote the Riemannian distance function on $M$. Let $p$ and $q$ be two points on $M$ and $X$ be a vector in $T_pM$. Let $C$ be the set of all the smooth paths defined on the unit interval joining $p$ to $q$. Define a metric $D:C\times C\to \mathbf R_{\geq 0}$ on $C$ as $$D(\gamma, \eta) = \sup_{t\in I} d(\gamma(t), \eta(t))$$

For $\gamma\in C$, let $P_\gamma:T_pM\to T_qM$ denote the parallel transport map corresponding to $\gamma$. I want to show the following:

The map $C\to T_qM$ defined as $\gamma\mapsto P_\gamma X$ is continuous.

In other words, if two paths in $C$ are nearby, then the resulting vectors got by parallel transporting along them gives two vectors which are also nearby.

I wanted to apply the result I am trying to prove here. If $\gamma$ and $\eta$ are two curves in $C$ which do not intersect and are contained in a single chart, then it seems intuitively obvious that the area enclosed between $\gamma$ and $\eta$ is small, and the result just alluded may be applicable. But of course, $\gamma$ and $\eta$ can intersect badly.