Universal quotient bundle need not have a nowhere-zero section
= Universal quotient bundle need not have a nowhere-zero section
Over $\operatorname{Gr}_1(\mathbb R^2)=\mathbb{RP}^1$, the universal quotient is the orthogonal-complement line bundle, a <Möbius line bundle>. Writing a section as $a(\theta)(-\sin\theta,\cos\theta)$ forces $a(\theta+\pi)=-a(\theta)$. Continuity then forces a zero, so no <nowhere-zero section> exists.