The heading is the question and here are my two approach, I want to know are they correct or not, if not I need to know the answer:
1) They are path connected as $\gamma(t)=At+(1-t)B, t\in [0,1]$ where $A$ and $B$ are trace one matrices. But I am not sure all matrices in this path are trace $1$?
2) as trace equals $1$, so considering diagonal entries I get a hyperplane $H$ with $x_{11}+\dots+x_{nn}=1$ and remaining other $n^2-n$ entries I can send to $\mathbb{R}^{n^2-n}$ and thus they are homeomorphic to $H\times \mathbb{R}^{n^2-n}$ as this is a product of two connected topological spaces, it is connected. So trace 1 matrices are connected. Well, here I have considered a matrix is just a point in $\mathbb{R}^{n^2}$
Thank you.
Best Answer
Partial answer for 1) $\text{tr} (\gamma(t))=t\cdot\text{tr}(A)+(1-t)\cdot\text{tr}(B)=t+(1-t)=1, \forall t$