Section 3.6 of Kobayashi and Nomizu's "Foundations of differential geometry, volume 1" says "A curve $\tau=x_t$, $a<t<b$, where $-\infty\leq a<b\leq\infty$, of class $C^1$ in a manifold $M$ with a linear connection is called a geodesic if the vector field $X=\dot{x}_t$ defined along $\tau$ is parallel along $\tau$, that is, if $\nabla_XX$ exists and equals 0 for all $t$, where $\dot{x}_t$ denotes the vector tangent to $\tau$ at $x_t.$"
I am aware that it is possible to define weak solutions of differential equations, and that there is research interest in considering geodesics of low regularity. However, how does Kobayashi and Nomizu's definition make sense with respect to the classical definitions? In such a setting, doesn't consideration of $\nabla_XX$ require the curve to be at least $C^2$, rather than $C^1$?