First of all, I have never taken a course about Lie algebra or Riemannian manifold, so please be kind about any of my inappropriate way of naming things or giving bad expression.
Suppose $n \in Z^+$, I am considering the space $\operatorname{SL}(n) = \{A \in R^{n \times n}, \operatorname{det} A = 1\}$. A equivalent relationship is defined on $\operatorname{SL}(n)$: $A \sim B$ if and only if there are similar to each other (i.e. there exists matrix $P$, such that $A = PBP^{-1}$.
I am wondering about what does $\operatorname{SL}(n) / \sim$ look like. Is this a quotient manifold?
Remark 1: I don't know about the exact way of describing the topology of $\operatorname{SL}(n)$, but I guess there should be some standard way, as $\operatorname{SL}(n)$ can be considered as an manifold in $R^{n \times n}$ in a natural way.
Remark 2: I have found some discussion (like What does this quotient space look like?), which considers a very similar problem but with the manifold $M_n(R)$ instead. In this case the quotient space might not even be Hausdorff. So I wonder the answer for the case where we exclude those bad(i.e. not invertible) matrices.
Remark 3: I'm not sure about the answer. On one hand the intuition tells me that the quotient space should be a manifold, because we already have excluded bad things; on the other hand, I think the answer is no because it seems to relate to the multiplicity of the eigenvalues.