Non-contractible closed geodesics on compact Finsler space forms without self-intersections

Let $M=S^n\mathrm{/}\mathrm{\Gamma }$ and $h\in {\pi }_1(M)$ be a non-trivial element of finite order $p$, where the integers $n,p\ge 2$ and $\mathrm{\Gamma }$ is a finite abelian group which acts on the sphere freely and isometrically, therefore $M$ is diffeomorphic to a compact space form which is typical a non-simply connected manifold. We prove there exist at least two non-contractible closed geodesics on $\mathbb{R}{P}^2$ and obtain the upper bounds on their lengths. Moreover, we prove there exist at least $n$ prime non-contractible simple closed geodesics on $(M,F)$ of prescribed class $[h]$, provided $$F^2<(\frac{\lambda +1}{\lambda }{)}^2{g}_0\text{ and }(\frac{\lambda }{\lambda +1}{)}^2<K\le 1\text{ for }n\text{ is odd or }0<K\le 1\text{ for }n\text{ is even},$$ where $\lambda$ is the reversibility, $K$ is the flag curvature and $g_0$ is standard Riemannian metric. Stability of these non-contractible closed geodesics is also studied.