1. Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that for every vector field $X$ on $M$ with a periodic orbit $\gamma$ and for every $s\in \Gamma^{\infty} (E)$, the section $\nabla_X s$ vanishes on at least one point of $\gamma$. Does this implies that $E$ is a (trivial) line bundle?

The motivation: The Rolle theorem is not valid in dimension greater than one.

2. Let $\ell$ be the canonical line bundle on $\mathbb{C}P^{1}\simeq S^2$, thought of as a smooth complex line bundle. Is there a connection $\nabla$ such that for every section $s$ and every periodic orbit $\gamma$ of $X$, the section $\nabla_{X}^{s}$ vanishes on at least one point of $\gamma$?

I ask the above question because I searched for some unusual differential operator associated with a vector field such that these operators can count the number of attractors of a vector field $X$. As a related post, please see: Elliptic operators corresponds to non vanishing vector fields