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.
2026-04-04 20:42:17.1775335337
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?
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