I am reading the John Lee's Introduction to Riemannian manifolds, 2nd Edition, proof of Theorem 12.28 ( Cheng's Maximal Diameter Theorem ) and stuck at final statement.
First, I will extract and ask main point. It seems easy ( works by unwinding definitions ? ) but can't prove rigorously until now.
Let $(M,g)$ be a complete, connected Riemannian $n$-manifold. And let $p \in M$ and $v\in T_p M$. Then the cut time $t_{cut}(p,v)$ and the injectivity domain $\operatorname{ID}(p)$ of $p$ are defined by
$$ t_{cut}(p,v):= \sup\{ b>0 : \gamma_v|_{[0,b]} \operatorname{is minimizing} \},$$
where $\gamma_v$ is the maximal geodesic starting at $p$ with initial velocity $v$. Because $\gamma_v$ is minimizing as long as its image stays inside a geodesic ball ( Prop. 6.11 ), $t_{cut}(p,v)$ is always positive, but it might be $+ \infty$.
$$ \operatorname{ID}(p):= \{ v\in T_pM : |v| < t_{cut}(p,v/|v|) \}$$
And the injectivity radius $\operatorname{inj}(p)$ of $M$ at $p$, is defined by
$$ \operatorname{inj}(p):=\sup\{a>0 : \exp_{p} \operatorname{is a diffeomorphim from} B_a(0) \subseteq T_p M \operatorname{onto its image} \}$$
Q. Then, my question is, if $\operatorname{inj}(p) = \pi R$ ( $ R>0$ ), then $\operatorname{ID}(p)$ is the entire ball of radius $\pi R$? If so, what should I catch?
This question originates from following proof ( John Lee's book Theorem 12.28 ) :
If needed, I will upload details in the proof. Why the final underlined statement is true? In the proof ( fourth paragraph ) we showed that $\operatorname{inj}(p_1) =\pi R$. And additionally in the underlined statement why John Lee mentioned that "on which the metric has the form (10.17) in normal coordinates " ? Where the sentence is used?