Minimizing properties of geodesics problem in do Carmo's book 2

Question

I'm reading DoCarmo's book Riemannian Geometry and in the section with minimizing properties of geodesics it this proposition.

At the final I don't understand why $r(1)=l(\gamma).$ Can some one fill in the details?

Answer 1

As $\gamma(1)=c(1)$ we know (by uniqueness) that \begin{equation} \gamma(t)=exp_p(tr(1)v(1)) \end{equation} and therefore, as $||v(1)||=1$ the curve $\gamma$ has lengt $r(1).$