The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal subgroup of $SU(k)$, so $SU(k)/SO(k)$ may not be a quotient group, or may not be any Lie group. However $SU(k)/SO(k)$ may be a manifold?
Question:
- IsIf $SU(k)/SO(k)$ is a manifold? for whichevery $k=?$$k$, how does this manifold behave?
- Can $SU(k)/SO(k)$ not be a manifold, for which $k=?$
- Are the different ways to specify the quotient so we may obtain different results? (see $k=2$ below)
Note that
$k=1$, $SU(1)/SO(1)= $ a point.
$k=2$, $SU(2)/SO(2)=SU(2)/U(1)= S^3/S^1= S^2$. However, there are different ways to have $S^1$ fibered over $S^2$, to get $S^3/\mathbf{Z}_k$. So I am interested in knowing $S^3/S^1$ can be something else other than $S^2$?
$k=3$, $SU(3)/SO(3)$ = Wu manifold as a 5 real dimensional manifold. But how exactly this is a manifold? How is this $SU(3)/SO(3)$ related to a Dold manifold as a $\mathbf{CP}^2$ fibered over $U(1)$? (correct me if I said the fibration the other way around.)
$k=4,\dots$, do we have a general understanding for this manifold?