I am sorry for too naive and stupid question, 


How can I express the 2nd Hirzebruch surface, $F\_{2}$ in terms of $SO(3)$. Can F\_{2} be realizable as the total space of a bundle over $\mathbb{R}\_{+}$ with fibre $SO(3)$. 


I have to admit that my knowledge of Hirzebruch surfaces is limited to the few lines in the wikipedia page. 



Hirzebruch surfaces
What you want almost works, but with a little twist. The group $SO(3)$ is acting on $F\_2$, so that there are two orbits that are $\mathbb CP^1$ and all other orbits are $SO(3)$. More precisely, $F\_2$ can be seen as a compactification of $\mathbb R\_+\times SO(3)$. Indeed, $F\_2$ can be seen as a compactificaton of $(\mathbb C^2\setminus 0)/\pm Id$, and $\mathbb C^2\setminus 0\cong \mathbb R\_+\times SU(2)$. Here $\mathbb R\_+$ means positive reals. 


**Details.** Each Hirzebruch surface $F\_n$ can be obtained by a quotient construction from $F\_1$. Recall, that $F\_1$ is the surface that is obtained by a blow up of a single point on $\mathbb CP^2$. I.e., we "replace" a point on $\mathbb CP^2$ by $\mathbb CP^1$. Of course we all know that $SU(2)$ is acting on $\mathbb CP^2$ fixing a point and a line at infinity (just take the usual action of $SU(2)$ on $\mathbb C^2$ fixing $0$ and extend it to the action on $\mathbb CP^2$). When we blow up $0$ on $\mathbb CP^2$ the action of $SU(2)$ extends on $F\_1$.


Now, $F\_n$ can be obtained from $F\_1$ by the quotient, $F\_n\cong F\_1/\mathbb Z\_n$, where $\mathbb Z\_n$ is the group of scalar matrices generated by $e^{2\pi i/n}\times Id$ acting in the standard way on $\mathbb C^2$ (and hence on $F\_1$). Finally, notice that in the case of $F\_2$ the corresponding group $\mathbb Z\_2$ belongs to $SU(2)$ (since $det(-Id)=1$). And since $SO(3)\cong SU(2)/\mathbb Z\_2$ this explains, hopefully, the construction. 


Of course all this is **completely** classical, but I can not give a reference from the top of my head.


