In the proof of Lemma 6.18 associated to the Hopf-Rinow theorem. I'm reading the John M.Lee, Introduction to Riemannian manifolds, second edition, p.167, Lemma 6.18 and I stuck at some statement :


My question is,
Question 1. Why can we write a unit-speed minimizing geodesic from $p$ to $q_i$(whose existence is gauranteed by the lemma) as form of
$\operatorname{exp}_p(tv_i)$ for some unit vector $v_i$?
Question 2. How can we prove the "$q_i=\operatorname{exp}_p(d_iv_i)$"?
I feel that I somewhat didn't understand about the exponential map.
Can anyone helps?
 A: 
Question 1. Why can we write a unit-speed minimizing geodesic from $p$ to $q_i$ (whose existence is guaranteed by the lemma) as form of $\exp_p(tv_i)$ for some unit vector $v_i$?

By the very definition of the exponential map, any geodesic $\gamma$ with initial data $\gamma(0)=p$, $\gamma'(0) = v$ can be written $\gamma(t) = \exp_p(tv)$ for all time $t$ such that $\gamma$ is defined. Moreover, any geodesic has constant speed, which is equal to $\|\gamma'(t)\|$ for any $t$, and in particular, it is equal to $\|\gamma'(0)\| = \|v\|$. If $\gamma$ has unit speed, then $\|v\|=1$.

Question 2. How can we prove the "$q_i=\exp_p(d_iv_i)$"?

For a minimizing geodesic, we have for all times $t$ and $s$, $d_g(\gamma(t),\gamma(s)) = |t-s|$. If $t\mapsto \gamma(t)=\exp_p(t v_i)$ is a unit speed minimizing geodesic joining $p$ to $q_i$ and if $t_0>0$ is such that $q_i = \exp_p(t_0v_i)$, then it holds that
$d_i=d_g(p,q_i) =d_q(\gamma(0),\gamma(t_0)) = |0-t_0| = t_0$.
A: Helped with the answer presented by Didier, I write an attempt to answer my question.

*

*The first question is true just because by his book Riemannian manifold (First Ed.), proposition 5.7-(b) : For each $V\in TM$, the maximal geodesic $\gamma_V$ is given by  $\gamma_V(t) = \operatorname{exp}(tV) $.


*I think that the second question also can be answered by next argument : Let $\gamma_ i(t) = \operatorname{exp}_p(tV_i) : [0,l] \to M$ be a unit speed minimizing geodesic  from    $p$ to $q$, where $l$ is the length of $\gamma_i$ (such existence of arc length parmetrization is gurranted by his book, p.93, Exercise 6.2 ).
Then, $$\operatorname{exp}_p(d_iV_i)=\gamma_{d_iV_i}(1)=\gamma_{V_i}(d_i)=\gamma_i(d_i)=\gamma_i(d(p,q_i))=\gamma_{i}(l)=q_i$$ For the first equality, we use the definition of the exponential map, for the second equality, we use the rescailing lemma(his book, p.73, Lemma 5.8), and for the previews equality before the last equality, we use that $\gamma_i$ is minimizing.
