The covariant acceleration formula is well-defined along a $C^2$ curve

Given: A smooth manifold $M$, an affine connection $\mathrm{\nabla }$, an interval $I\subseteq \mathbb{R}$ with nonempty interior, and a curve $z:I\to M$ that is locally $C^2$ in the chartwise sense of Covariant acceleration along a $C^2$ curve, including its one-sided endpoint conditions.

Covariant acceleration along a $C^2$ curve: in a chart $x$, $U^a=\frac{d}{d\tau }(x^a\circ z)$ and $a_x^a={\dot{U}}^a+{\mathrm{\Gamma }}^a{}_{bc}{U}^b{U}^c$ is the candidate acceleration component; $z$ is locally $C^2$ in chart coordinates, with first and second derivatives extending continuously and one-sidedly to included endpoints.

Connection one form in a local frame and Christoffel symbols of an affine connection: in the coordinate frame, the connection matrix has entries ${\omega }^a{}_b={\mathrm{\Gamma }}^a{}_{cb}d{x}^c$, and a section with component column $u$ has covariant derivative $du+\omega u$.

Connection one form transformation law: if two frames satisfy $e^{\mathrm{\prime }}=eA$, then their connection matrices obey ${\omega }^{\mathrm{\prime }}=A^{-1}\omega A+A^{-1}dA$.

$C^r$ and smooth maps between smooth manifolds and $C^k$ Euclidean maps are closed under componentwise algebra and composition: chart transition maps are smooth, and their compositions with the locally $C^2$ curve have $C^2$ coordinate representatives. On the interior this follows by composition; at an included endpoint the first- and second-derivative chain-rule expressions have continuous one-sided limits because the transition map is smooth and the curve derivatives through order two do.

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length and Covariant acceleration along a $C^2$ curve: $I$ is order-convex with nonempty interior, and the local $C^2$ definition supplies continuous derivatives through order two and one-sided endpoint values at included endpoints.