Suppose $M$ a manifold with dimension $m$ and $E$ a vector bundle whose based manifold is $M$.
Assume $\mathrm{dim} E$ is $2m+1$, by Transversality theorem, seems I can obtain a global section without zero point, because we have a transversal map $s:M \to E$ but $2\times \mathrm{dim}M <\mathrm{dim}E$, so $s$ will not intersect to the zero section.
It's surprising, do I misunderstand something? If it's true, is there any natural concept to explain it? Thanks!