Let $G$ be a compact Lie group.



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> Is each conjugacy class a closed subset of $G$?
>  Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with the quotient topology?Is a there a natural manifold structure on it? If the answer is "yes", is there a natural Riemannian meyric on it such that the the quotient map would be a partial isometry?(After we fix a left invariant metric on $G$)
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A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group
After you choose a maximal torus $T$, the space of conjugacy classes is identified with with the quotient $T / W(T)$ of $T$ by the Weyl group, see <https://en.wikipedia.org/wiki/Maximal_torus#Weyl_group>. The quotient is not a manifold in general (if $G$ is simply connected, then it can be identified with a Weyl chamber).


