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Let $M=S^n/ \Gamma$ and $h \in \pi_1(M)$ be a non-trivial element of finiteorder $p$, where the integers $n, p\geq2$ and $\Gamma$ is a finite abeliangroup which acts on the sphere freely and isometrically, therefore $M$ isdiffeomorphic to a compact space form which is typical a non-simply connectedmanifold. We prove there exist at least two non-contractible closed geodesicson $\mathbb{R}P^2$ and obtain the upper bounds on their lengths. Moreover, weprove there exist at least $n$ prime non-contractible simple closed geodesicson $(M,F)$ of prescribed class $[h]$, provided \[ F^2<(\frac{\lambda+1}{\lambda})^2 g_0 \;\; \text{ and } \;\;(\frac{\lambda}{\lambda+1})^2 < K \leq 1 \text{ for $n$ is odd or }\; 0<K \leq1 \text{ for $n$ is even}, \] where $\lambda$ is the reversibility, $K$ is theflag curvature and $g_0$ is standard Riemannian metric. Stability of thesenon-contractible closed geodesics is also studied.

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