in all questions $(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature

1. let $\alpha$ be a simple closed geodesic loop in $M$

if $n=2$ then we know that the geodesic in the free homotopy class of $\alpha$ is simple . now suppose $n>2$ will the geodesic in the free homotopy class of alpha be simple ?

2. suppose $n>3$ and let $\alpha$ be a simple closed geodesic loop at a point $p$ in $M$ can $\alpha$ be homotopic ( with respect to $p$ ) to a power of another closed curve at $p$

3. suppose $n>3$ let $p$ be a point in $M$ . Let $\Gamma$ be subgroup of $\pi_1(M,p)$ generated by {$\alpha \in \pi_1(M,p)$ such that $ length(\alpha)= Systole(M,p)$} . suppose now $\Gamma$ is cyclic does that necessary mean that there exist a unique $\alpha$ such that $length(\alpha) = Systole(M,p)$

$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$

1. will the geodesic in the free homotopy class of alpha be simple ?

2. can $\alpha$ be homotopic ( with respect to $p$ ) to a power of another closed curve at $p$