Let $E\to M$ be a real smooth vector bundle and $N_{1},N_{2},\ldots,N_{k}$ are disjoint compact (proper) submanifolds of $M$. Are there smooth sections $s_{1},s_{2},\ldots,s_{k}$ such that for every nontrivial linear combination $s=\sum \lambda_{i} s_{i}$ there exist an index $j\in \{1,2,\ldots,k\}$ such that $s$ is non vanishing on $N_{j}$? Note that this is true when $E$ possess a global non vanishing section.
This question is indirectly related to The "Rolle theorem" for sections of a vector bundle
