I'm a physicist studying control problems in quantum theory, and one of the ways to understand this is to consider coset spaces and Lie groups. What I am curious about are the manifolds associated with various quotient groups, and I am interested in two specific cases.
The first arises from a Cartan decomposition of a Lie algebra $\cal{L}$ into $\cal{L} = \cal{K} + \cal{P}$, where $\cal{P} = \cal{K}^{\perp}$. If the Lie algebra is $su(n)$ we can perform a Cartan decomposition $su(n) = so(n) \oplus {\cal{I}}$ where ${\cal{I}}$ is the vector space in $su(n)$ of purely imaginary matrices. Understanding the associated control problem involves studying the quotient space $e^{\cal{L}}/e^{\cal{K}}$.
This means I need to understand the topology of the quotient space $G/K$, where $G = SU(n)$ and $K=SO(n)$.
Question 1: I know $SU(n)/SO(n)$ should be a differentiable manifold of dimension $\frac{1}{2}n(n+1)-1$, but what exactly is it? A hypersphere? A hypertorus? As an example, I know the answer for $n=2$: the quotient space $SU(2)/SO(2)$ is isomorphic to the two-sphere $S^2$. But I can't find the answer for $n>2$.
Question 2: Exactly the same as above, but now I don't have a Cartan decomposition and the quotient group manifold I need to find is given by $SU(n)/U(1)$.
Any suggestions gratefully accepted. Since I'm new to this, and I'm a physicist and not a mathematician, it's possible my questions don't even make sense, in which case I would appreciate anyone pointing out my misconceptions!