I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408.2760, $ SO(2n+1)/U(n) $ is not a symmetric space because it does not have the right Cartan decomposition of the Lie algebra. That is, suppose that $ \mathfrak{g} $ is the Lie algebra of $ SO(2n+1)$ and $ \mathfrak{h} $ is the Lie algebra of $ U(n) $. There is a decomposition $$ \mathfrak{g} = \mathfrak{h} + \mathfrak{p} $$ for some $ \mathfrak{p} $ such that $ [\mathfrak{h},\mathfrak{h}] \subset \mathfrak{h}$ and $[\mathfrak{h},\mathfrak{p}] \subset \mathfrak{p}$. But $ [\mathfrak{p},\mathfrak{p}] $ is not in $ \mathfrak{h} $, so we do not have a Cartan involution on this space.
I'm wondering if it is that simple. I'll be grateful if someone can clear up my confusion below.
I think that $SO(2n+1)/U(n)$ and $SO(2n+2)/U(n+1)$ are diffeomorphic. For instance, this book shows that the two are the same homogeneous spaces by showing that
- Any element $SO(2n+1)$ can be written as an ordered product of two elements, one in $SO(2n+1)$ and another in $U(n+1)$. They write this as $SO(2n+2)=SO(2n+1)\cdot U(n+1)$
- The quotient $X = SO(2n+2)/U(n+1) = SO(2n+1)\cdot U(n+1) / U(n+1)$ can be thought of as $ SO(2n+1)/U(n)$ because the $SO(2n+1)$-action on $X$ is transitive and the stabilizer of the identity $eU(n+1)$ of $X$ are the elements of $SO(2n+1)$ that are also in $U(n+1)$: $SO(2n+1) \cap U(n+1) = U(n)$.
An example in low dimensions is $ SO(5)/U(2) = SO(6)/U(3) = \mathbb{C}P^3$, a complex projective space.
But $SO(2n+2)/U(n+1)$ is a symmetric space. So why is $SO(2n+1)/U(n)$ not?
