I have a problem to identify $\mathbb{R}\mathbb{P}^3$ with the spherical tangent bundle of $S^2$, $ST(S^2)\cong SO(3)$. The spherical tangent bundle is just a sub bundle of the fiber bundle consisting of vectors of norm 1.
The book I am reading takes a ball $B^2\subset \mathbb{R}^3$ of radius $\pi$. For each point $x$ of $B^3$ we can associate the rotation of axis $ox$ and angle $|x|$, and since antipodal points of $\partial B^3$ rotate $\mathbb{R}^3$ around the same axis and with angle $\pi$, they define the same rotation of $\mathbb{R}^3$, so we can identify them.
Finally, the author said that this is a $3$-manifold obtained from this ball by indentifying antipodal points of its boundary.
It is probably a stupid question, and sorry for that, but, why isn't it $\mathbb{R}\mathbb{P}^2$ instead of $\mathbb{R}\mathbb{P}^3$?