By Lemma 5.2 for any $t\in[0,T]$, the minimizing geodesic $\gamma$ connecting $p_{1}(t)$ and $p_{2}(t)$ lies in $Y^{reg}$. So there is an $\epsilon>0$ so that the curve $\gamma_{s}=\exp(s\xi).\gamma$ is in $Y^{reg}$ for $s\in[0,\epsilon]$. Clearly the length of $\gamma_{s}$ is independent of $s$. Thus $f(t)$ is a decreasing function. Replace $\xi$ by $-\xi$ one sees that $f$ is also an increasing function. Thus $f$ is constant.
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