The "Rolle theorem" for sections of a vector bundle 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
 A: As a first observation, if $(E,\nabla)$ has this Rolle property then if $i: S^1 \to M$ is any embbeding, $(i^*E,i^*\nabla)$ has this property as well (this is just a tubular neighborhood + cutoff argument). In particular we need to understand what possible such vector bundles with connection appear over $S^1$. It is fairly clear that the only possibilities are line bundles. If $i^*E$ is the Mobius bundle, then any connection has this property. If it is the trivial bundle, then the connection must have trivial holonomy (in other words there must be a trivialization for which the connection $i^*\nabla$ is the canonical one associated to that trivialization). In particular the connection must be flat.
For $\dim M > 2$ this classifies all such bundles rather satisfactorily: they are precisely the line bundles with a metric, equipped with the unique connection that preserves this metric. This is because any embedded circle in $\dim \geq 3$ admits a family of embeddings $i_n:S^1 \to M$ converging to the immersion of multiplicity two $z \mapsto i(z^2)$. The holonomy along these converges to the holonomy along the multiplicity two embedding, so the holonomy along $i$ must be $-1$ (a priori it could have been any negative scaling). 
In dimension two, the above argument breaks down, and indeed there are line bundles on $S^1 \times \mathbb{R}$ that have holonomy a negative number $\neq 1$. However, there is clearly some reasonable casework that could be done: on $\mathbb R P^2$ all line bundles satisfying the condition are metric compatible: all homotopically nontrivial embedded circles have nontrivial normal bundle, so we may find embeddings $i_n$ converging to $i$ with multiplicity two.  In the torus, we simply see that there are no holonomy maps $\mathbb{Z}^2 \to \mathbb{R}^{\times}$ which take on values besides $\{ \pm 1\}$ but also take on negative values or $1$ at every class representable by an embedding (namely if some class takes on a value beside $\pm 1$, some class representing an embedding takes on a positive value not equal to $1$), so again all line bundles satisfying our condition are metric compatible. (this section was edited)
