Let $\mathfrak S(n)$ be the set of all positive definite $n\times n$ (real) matrices. I am no expert in topology but stumbled across this question: Does there exist a smooth path through $\mathfrak S(n)$?
I know already that $\mathfrak S(n)$ can be made to a topological space by equipping the set with the topology inherited from the canonical inner product for $n\times n$ matrices $A$, $\Vert A\Vert = \sqrt{\mathrm{trace}(A'A)}$. But what are the requirements for a smooth path to exist?
For any $A, B \in {\mathfrak S}_n$, $(1-t) A + t B$, $0 \le t \le 1$, is a smooth path in ${\mathfrak S}_n$ from $A$ to $B$.