Let $(T^n, g_0)$ be the flat $n$-torus. For every $c>0$, it is known that a small $C^{\infty}$-perturbation of $g_0$ produces a metric $g$ such that all closed geodesics of length less or equal to $c$
are non-degenerated as critical points of the energy functional on the loop space;
and have Morse indices less or equal to $n-1$.
Is it true that for every contractible open set $U\subset T^n$ and $c=\pi$, a small $C^{\infty}$-perturbation of $g_0$ produces a metric $g$ with the two properties above with the additional property that every closed geodesic of length less or equal $\pi$ intersects $U$?
