Universal quotient bundle on a real Grassmannian ($Q_W={\mathbb{R}}^N/W$)

Over ${\mathrm{Gr}}_k({\mathbb{R}}^N)$, quotient the trivial bundle by the tautological bundle. The quotient is a vector bundle because it identifies continuously with the orthogonal-complement bundle via $[v]\mapsto P_{W^{\perp }}v$. Its rank is $N-k$.

Over ${\mathrm{Gr}}_1({\mathbb{R}}^2)={\mathbb{RP}}^1$, 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.

On a compact base, finitely many global sections can be selected to span every fiber when evaluations of global sections are surjective. They give a surjective morphism $M\times {\mathbb{R}}^N\to E$. Its kernel is a kernel bundle of a surjective vector bundle morphism, and its fibers give the displayed map $\varphi$. The first isomorphism theorem on each fiber identifies the resulting pullback quotient with $E$, with continuously varying inverses. The finite selection follows by choosing a frame's values at each point, extending each value to a global section, then taking a finite cover of the regions where those sections remain independent.