It is well known that the Grassmanian of lines in $\mathbb P^3$ is isomorphic 
to a quadric in $\mathbb P^5$. I would like to ask if the tautological rank two bundle on the grassmanian extends to a rank two bundle on the whole of $\mathbb P^5$?



Extending the tautological bundle of $G(1,3)$?
I would say it doesn't. The tautological Chern classes generate $H^\*$ of the Grassmanian, but the inclusion to $\mathbb{P}^4$ does not induce an epimorphism.


