Let $\gamma : [ 0, 1 ] \rightarrow \mathcal M$ and $\sigma : [ 0, 1 ] \rightarrow \mathcal M$ be two paths,
such that both have the same image, i.e. the curve $\gamma[ \, [ 0, 1 ] \, ] \equiv \sigma[ \, [ 0, 1 ] \, ] \subset \mathcal M$,
and such that the compositions $\sigma^{(-1)} \circ \gamma : [ 0, 1 ] \rightarrow [ 0, 1 ]$ as well as $\gamma^{(-1)} \circ \sigma : [ 0, 1 ] \rightarrow [ 0, 1 ]$ both exist and are both differentiable.
Thus $\sigma$ is a differentiable reparametrization of $\gamma$, and vice versa.
Is consequently $$ \nabla_{\left(\frac{\dot\gamma}{\|\dot\gamma\|}\right)}\left[\,\frac{\dot\gamma}{\|\dot\gamma\|}\,\right]_{(t \, := \gamma^{(-1)}[ \, P \, ])} = \nabla_{\left(\frac{\dot\sigma}{\|\dot\sigma\|}\right)}\left[\,\frac{\dot\sigma}{\|\dot\sigma\|}\,\right]_{(t \, := \sigma^{(-1)}[ \, P \, ])} := \vec a[ \, P \, ], $$ for any $P \in \mathcal M$, any affine connection $\nabla$
and for any (generalized) norm $\| v \|$$:= \text{sgn}[ \, g[ \, v, v \, ]\, ] \sqrt{ g[ \, v, v\, ] \text{sgn}[ \, g[ \, v, v\, ] \, ] }$
?
Does the value $\vec a[ \, P \, ]$ have an immediate geometric interpretation, perhaps as the product $(\kappa \, {\mathsf N})_P$ of curvature $\kappa$ and normal unit vector $\mathsf N$ of curve $\gamma[ \, [ 0, 1 ] \, ]$ at point $P$ ?