Parameter space of $SO(3)$ and $SU(2)$

Question

1. Is it parameter space of $SO(3)$ and $SU(2)$ are same?

2. can we use quaternions to represent both groups?

3. what about their connectedness?

Answer 1

The group manifold of ${\rm SU}(2)$ is the three sphere $S^3$. The group manifold of ${\rm SO}(3)$ is the three-sphere with antipodal points identified. The two spaces have different connectectness (measured by homotopy) because $\pi_1({\rm SU}(2))= \{0\}$ and $\pi_1({\rm SO}(3)={\mathbb Z}_2$. ${\rm SU}(2)$ is a double cover of ${\rm SO}(3)$. You can identify ${\rm SU}(2)$ with the group of unit length quaternions by the homomorphism $$ U= x_0{\mathbb I}-ix_1 \sigma_1-ix_2 \sigma_2-ix_3 \sigma_3 \leftrightarrow {\bf q}=x_0+x_1{\bf i}+x_2{\bf j}+x_3 {\bf k}. $$ Here $U\in {\rm SU}(2)$ and ${\bf q}\in {\mathbb H}$ and $x_0^2+x_1^2+x_2^2+x_3^2=1$. To get ${\rm SO}(3)$ you identify ${\bf q}$ with $-{\bf q}$.