How do you prove that the universal covering of $SO(3, \mathbb{R})$ is $S^3$ ? Or equivalently, that it is diffeomorphic to $P_3\mathbb{R}$ ?
Thank you for your answers.
How do you prove that the universal covering of $SO(3, \mathbb{R})$ is $S^3$ ? Or equivalently, that it is diffeomorphic to $P_3\mathbb{R}$ ?
Thank you for your answers.
Any element of $SO(3)$ is rotation about an axis in $\mathbb{R}^3$ - that is each element can be represented by an axis of rotation and an angle of rotation.
$\mathbb{RP}^3$ is $\mathbb{D}^3$ with antipodal points identified. Given a point in $\mathbb{D}^3$ it is some distance (between -1 and 1) on a vector from the origin. This vector gives you the axis of rotation for a point in $\mathbb{RP}^3$. We still need the angles of rotation, but these will be given by the distance between -1 to 1. Scale those values to be $-\pi$ to $\pi$, Note that then antipodal points of $\mathbb{D}^3$ are mapped to the same point in $\mathbb{RP}^3$
One can also show that the adjoint representation of su(2) is so(3) and find the covering map quite explicitly.