Chapter 3, section 2.
Definition 2.1. : A parametrized curve $\gamma : I \to M$ is a geodesic at $t_0 \in I$ if $\frac{D}{dt} \left( \frac{d\gamma}{dt}\right) = 0$ at the point $t_0$; if $\gamma$ is a geodesic at $t$, for all $t \in I$, we say that $\gamma$ is a geodesic. If $[a,b] \in I$ and $\gamma : I \to M$ is a geodesic, the restriction of $\gamma$ to $[a,b]$ is called geodesic segment joining $\gamma(a)$ to $\gamma(b)$.
Right after this definition the following expression is given for the arc length
$$ s(t) = \int_{t_0}^t \left| \frac{d \gamma}{dt}\right| dt = c(t - t_0) $$ with the assumption $c = \left| \frac{d \gamma}{dt}\right|$ is some non zero constant.
I don't recall Do-Carmo introduced the notation $\left| \cdot \right|$ anywhere but I would assume it should be interpreted as
$$ \sqrt{\left\langle \frac{d \gamma}{dt} , \frac{d \gamma}{dt} \right\rangle}_{\gamma(t)} = \left| \frac{d \gamma}{dt}\right| $$
Where $\left\langle \cdot, \cdot \right\rangle_p$ Is the Riemannian metric defined on the manifold $M$, am I right?
Yes, you are right. This is standard notation for the Euclidean length of a vector.