Let $M$ be a complete connected Riemannian manifold and $r\in M$ and $p,q\in B(r;R)$ where $R$ is the convexity radius at point $r$. Let $u\in T_pM$ and $v\in T_qM$ and $\epsilon>0$ be such that $\exp_psu,\exp_qsv\in B(r;R)$ and $\Gamma:(-\epsilon,\epsilon)\times[0,1]\to M$ be defined by
$$\Gamma(s,t)=\exp_{\exp_psu}(t\exp^{-1}_{\exp_psu}\exp_qsv)$$
I want to calculate the variation field $V(t)=\partial_s\Gamma(0,t)$ at $t=0$ and $t=1$, that is
$$\partial_s\Gamma(0,t)=\frac d{ds}\Gamma(s,t)|_{s=0}$$
If $\gamma(s)=t\exp^{-1}_{\exp_psu}\exp_qsv$ ,then $$\frac d{ds}\Gamma(s,t)|_{s=0}=(\exp_{\exp_psu})_{*t\exp^{-1}_pq}(\dot\gamma(0))$$
if $\mu=\exp_qsv$, then $$\dot\gamma(0)=(t\exp^{-1}_{\exp_psu})_{*q}(\dot\mu(0))$$ So, $$V(0)=(\exp_{\exp_psu})_{*0}((0)_{*q}(\dot\mu(0)))$$ and $$V(1)=(\exp_{\exp_psu})_{*\exp^{-1}_pq}((\exp^{-1}_{\exp_psu})_{*q}(\dot\mu(0)))$$ but I don't know how continue from here. In this paper is claimed that $V(0)=u$ and $V(1)=v$ without any explanation. Can anyone give me some hint for calculating of $V(0)$ and $V(1)$, please?
To calculate $V(t)$ at $t = 0$, we need to calculate the partial derivative $$\frac{\partial}{\partial s} \Gamma(t,s)|_{t=0,s=0} = \frac{d}{ds} \Gamma(0,s)|_{s = 0}.$$
Note that we have
$$ \Gamma(0,s) = \exp_{\exp_p(su)}(0) = \exp_p(su)$$
so $\Gamma(0,s)$ is a geodesic eminating from $p$ in the direction $u$. Hence,
$$ \frac{d}{ds} \Gamma(0,s)|_{s = 0} = \frac{d}{ds} \exp_p(su)|_{s = 0} = u. $$
Similarly,
$$ \Gamma(1,s) = \exp_{\exp_p(su)} \left( \exp_{\exp_p(su)}^{-1}(\exp_q(sv))\right) = \exp_q(sv)$$
so again, $\Gamma(1,s)$ is a geodesic eminating from $q$ in the direction $v$ and so $\frac{d}{ds} \Gamma(1,s)|_{s = 0} = v$.