In the mentionned paper, I am having difficulties in understanding the proof of lemma 2. Roughly, this lemma says that given any separation $\eta$ for the $C^\infty$ topology of smooth paths from $[-1,1]$ to the manifold $M$ (we assume a metric as been chosen, as this is metrizable), given any point $x\in M$ and a compact neighborhood $K$ of it, we can find a radius $\epsilon$ for the tangent spaces in $K$ such that any unit speed geodesic starting from $K$ and some kind of parallel curve (not necessarily geodesic) to it made of a vector of length small than $\epsilon$ will stay within an $\eta$ distance of the geodesic.
To put things in context, and adapting the notations, we have the following :
Given a unit tangent vector $v$, let $\gamma_v$ be the geodesic starting with velocity $v$. Given a small enough tangent vector $w$ at the same point as $v$, we can define a curve which I denote $\gamma_{v\parallel w}$ on a small time interval as follows : parallel translate $w$ along $\gamma_v$ for a time t, then exponentiate this vector.
Lemma 1 says that given a positive number $\eta$ and $v$ unit speed, we can find an $\epsilon$ (depending on $v$) such that the "parallels" (in the above sense) made of vectors $w$ of length $\leq\epsilon$ stay at distance $<\eta$ of $\gamma_v$.
The point of the proof is that if $B$ is the unit closed ball in $T_xM$, then we can consider the map $F:B\times [-1,1] \rightarrow C^\infty ([-1,1],M) : (w,s) \mapsto \gamma_{v\parallel sw}$ and find a neighborhood of $B\times {0}$ in $B\times [-1,1]$ sent to within a distance $\eta$ of $\gamma_v$.
Lemma 2 says we can find an $\epsilon$ which wouldn't depend on the unit vectors in a compact neighborhood $K$.
The argument given there, which I don't understand, is : given $\epsilon$ which satisfies lemma 1 for the given $\eta$ and some point $p$ in $K$ and some unit speed $v$, then $\epsilon/2$ satisfies lemma 1 for all $(p',v')$ with $p'\in K, v'\in T_xM$ of length 1 and $(p',v')$ sufficiently close to $(p,v)$ in $TM$.
The authors claim this follows from the continuous dependence of geodesics and the exp map.This last point is of course true, but I don't precisely see how this shows the claim (it seems like everything could be true from such arguments...).
I am also perplexed at why not use an argument similar to lemma 1, namely use the following map : if $S$ is the unit sphere bundle in $TM$ and $D$ the closed unit ball bundle in $TM$, then consider the map $(v,w,s)\mapsto \gamma_{v\parallel sw}$ defined for $v\in S$, $w\in D$ both at the same point in $K$ and $s\in [-1,1]$. Then consider $L$ the compact subset of $C^\infty ([-1,1],M)$ made of the the unit speed geodesics starting from points in $K$, and the open neighboorhood made of paths that are at distance less than $\eta$ from $L$.
Thank you for any help.
