# Circle tangent bundle over $S^{2}$

Let $S_{r}^{2}$ be a sphere of radius $r$ and let $TS_{r}^{2}$ be its tangent bundle. If

$SS^{2} = \{(x,p)\in TS_{r}^{2}| |p|=1 \}$ be the circle tangent bundle of non zero radius . Then are there any descriptions of $\mathbb{R}_{+} \times SS^{2}$ in terms of known manifolds. e.g. It seems that (correct me if I'm wrong)

$\mathbb{R}_{+} \times SS^{2} \simeq TS_{r}^{2} - S_{r}^{2}$ via the map $(a,x,p) \mapsto (p/a,ax)$.

Also,from the answers to this questions one can infer that $\mathbb{R}_{+} \times SS^{2} \simeq \mathbb{R}_{+} \times SO(3)$?

Are these two descriptions same, and are there any other such descriptions.

Can we somehow classify such spaces,I don't know how to make this precise. It seems like these are all circle bundles over $\mathbb{R}_{+} \times S_{r}^{2}$ tangent to the sphere.

I am sorry for my naivety and not a well posed questions. Thanks.

• By your definition of $SS^2$ we immediately have $SS^2=TS^2_r-S_r^2$. Maybe you wanted to define $SS^2:=\{(x,p)\in TS^2_r\,|\, |p|=1\}$? – Dave Mar 4 '13 at 16:06
• @DaveHartman yes! Thanks for pointing. Now edited. – J Verma Mar 5 '13 at 4:03
• what do you mean for: It seems like these are all circle bundles over ${\Bbb{R}}_+\times S^2_r$ tangent to the sphere"...?? – janmarqz Jan 12 '14 at 19:36