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Ali Taghavi
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A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural embedding$([x],v)\mapsto [x,x.v]$):

http://math.stackexchange.com/questions/1486105/is-the-total-space-of-the-tautological-line-bundle-over-mathbbrpn-a-non

Now assume that $E$ is the total space of the tautological 2-plane bundle over the real Grassmanian $G(2,n)$ .As a generalization of the above fact we ask:

Is there a (natural) embedding of $E$ into $G(2,n+1)$? If yes, what is the topological type of the remainder $E\setminus G(2,n+1)$. Is it possible to have an embedding with a one point remainder?

Ali Taghavi
  • 356
  • 8
  • 31
  • 123