Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in$\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which actsfreely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphicto a compact space form. In this paper, we establish first the resonanceidentity for non-contractible homologically visible minimal closed geodesics ofthe class $[h]$ on every Finsler compact space form $(M, F)$ when there existonly finitely many distinct non-contractible closed geodesics of the class$[h]$ on $(M, F)$. Then as an application of this resonance identity, we provethe existence of at least two distinct non-contractible closed geodesics of theclass $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result ofTaimanov in [Taimanov 2016] by removing some additional conditions. Also ourresults extend the resonance identity and multiplicity results on$\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms.
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机译：令$ M = S ^ n / \ Gamma $和$ h $是有限阶$ p $ in $ \ pi_1（M）$的非平凡元素，其中整数$ n \ geq2 $，$ \ Gamma $是有限元该组在$ n $球面上自由地，等距地作用，因此$ M $会微变为紧凑的空间形式。在本文中，我们首先建立了在$（M，F）$的每个Finsler紧空间上，当仅存在有限的许多不同的非可收缩封闭测地线时，在类[[h] $ $（M，F）$上的类$ [h] $。然后，作为该共振身份的一种应用，我们证明了存在至少两个截然不同的不可收缩的闭合测地线，它们具有颠簸的Finsler度量，在（M，F）$上的$ [h] $类，从而改善了Taimanov在[泰马诺夫[2016年]。同样，我们的结果将[arXiv：1607.02746]中的\ mathcal {R} P ^ n $的共振身份和多重性结果扩展到了一般的紧凑空间形式。
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