Geodesic-unclear operator $\nabla$

Question

I understand what is a connection $$\nabla_X\textbf{v}$$ for a vector $X$ and a vector field $\textbf{v}$ near point $p$.

However I do not understand $(10.5)$ below, namely what is $$\frac{\nabla}{dt}$$

The snippet:

Answer 1

This stands for the Levi-Civita connection along the curve $\mathbf{x}(t)$. By definition it is $$ \frac{\nabla}{dt}\left(\frac{d\mathbf{x}}{dt}\right) = \nabla_{\frac{d\mathbf{x}}{dt}} X $$ where $X$ is a local vector field on $M$ such that $X(\mathbf{x}(t)) = \frac{d\mathbf{x}}{dt}(t)$. This definition makes sense because the value $\nabla_X Y(p)$ only depend on the vector $v=X(p)$ at the point $p$ and the values of $Y$ along a curve $\alpha:I\to M$ with $\alpha(0)=p$ and $\alpha'(0) = v$.

Intuitively one can course just think of this as "the Levi-Civita connection restricted to the curve". However, there are some technicalities involved, and in order to emphasize these, some references use a separate notation, such as $\nabla/{dt}$ or $D/dt$. Other sources use the more loose notation $\nabla_{d\mathbf{x}/dt}$.