# Is there a relationship or connection between compact groups like $U(n)$ and manifolds like $S^{n-1}$ (unit sphere)? If so, what is it?

I am interested in simulating a random walk on unitary groups using a computer and am wondering if it bears any similarity to a random walk on a recognizable manifold like the unit sphere in some number of dimensions. More generally, I'm wondering if the group is isomorphic or in some way similar / analogous / related to such a manifold or geometric object.

• You are aware that the group is itself a manifold? Apr 24 '17 at 21:41
• @MattSamuel, no. This is why I asked. References describing this would be helpful, as well as any useful connection to other manifolds (for example, something that helps a type of visual intuition, or something that helps with computation). Apr 24 '17 at 21:46
• I'm sure there's a book that discusses the geometry of the unitary group, but I don't know of one. In the meantime you can look up Lie groups. It turns out it's very easy for a topological group to be a manifold. If it's not a manifold (or the infinite dimensional equivalent) then it has horrible topological properties. Apr 24 '17 at 22:47