Vector field and section along a smooth curve

Let $\pi :E\to M$ be a smooth vector bundle. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior and let $\gamma :I\to M$ be smooth, with smooth local extensions at any included endpoints. Give ${\gamma }^{\ast }E=\{(t,e)\in I\times E:\gamma (t)=\pi (e)\}$ the subspace topology. For a bundle chart $\mathrm{\Phi }:E{\mathrm{\mid }}_U\to U\times {\mathbb{R}}^r$, writing $\mathrm{\Phi }(e)=(\pi (e),v)$, the pulled-back chart and its inverse are $(t,e)⟼(t,v),(t,v)⟼(t,{\mathrm{\Phi }}^{-1}(\gamma (t),v))\mathrm{.}$ They are continuous for the subspace and product topologies, and the overlap maps are $(t,v)\mapsto (t,g_{\beta \alpha }(\gamma (t))v)$, hence smooth and fibrewise linear. Thus they directly define a smooth rank-$r$ bundle over $I$, with the stated smooth-up-to-endpoint convention. The interval $I$, as a subspace of $\mathbb{R}$, and the manifold $E$ are Hausdorff and second countable. Their product is Hausdorff by Arbitrary products preserve $T_0$, $T_1$, and Hausdorffness, and products of their two fixed countable bases form a countable basis. Hence both properties pass to the subspace ${\gamma }^{\ast }E$ by $T_0$, $T_1$, and Hausdorffness are hereditary and Second countability is hereditary.

A section of $E$ along $\gamma$ is a smooth section of this bundle ${\gamma }^{\ast }E$. Equivalently it is a smooth map $V:I\to E$ with $\pi (V(t))=\gamma (t)$. For $E=TM$ it is a vector field along $\gamma$. In a pulled-back frame it has the form $V(t)={\sum }_j{v}^j(t)e_j(\gamma (t))$ with arbitrary smooth coefficients $v^j$.

Values belong to the fibre over the parameter value, even if $\gamma (t)=\gamma (u)$ for $t\mathrm{\ne }u$. For example, for a constant curve at $p$ and $v\mathrm{\ne }0$ in $E_p$, the section $V(t)=tv$ is allowed and cannot be $s\circ \gamma$ for an ambient section $s$. Thus an ambient extension is not part of this definition.

For a piecewise smooth curve on a compact interval, use a finite subdivision into smooth pieces (each smooth up to its endpoints). A piecewise smooth section is continuous on the whole interval and smooth on each piece. On a singleton interval it is just a fibre vector; on an empty interval it is the empty section. These conventions define sections, without attempting differentiation on a singleton.