Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group $U(N)$.
Now consider a Lie subalgebra $g$ of $u(N)$; $g$ is a proper subalgebra (non trivial, but not $u(N)$). Let $a \in u(N)$ be fixed, then $a + g$ is an affine subspace which I call `affine subalgebra' (I realize that this may not be a subalgebra of $u(N)$).
How does one think about $exp(a + g) =\{exp(a + y) | y \in g\} $? I guess that this will be some submanifold of $U(N)$ (viewed as a manifold)...will this be a symmetric space?