We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right).$ Then ${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and ${\sf crl}\left(M,g\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik and Fet in the general case. It is the main result of the paper that the numbers ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere. Under the additional assumption $K \le 4$ this result for ${\sf crl}\left(M,g\right)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson and Ziller in odd dimensions.

中文翻译：

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环同调类和正曲率的临界值

我们研究具有正截面曲率 $K\ge 1.$ 的紧凑且简单连接的黎曼流形。分别定义一个临界长度 ${\sf crl}_p\left(M,g\right)$。${\sf crl}\left(M,g\right).$ 则 ${\sf crl}_p\left(M,g\right)$ 等于测地线环的长度，${\sf crl}\ left(M,g\right)$ 等于闭合测地线的长度。这是 Birkhoff 在球体的情况下和 Lusternik 和 Fet 在一般情况下提出的正长度闭合测地线存在的证明的想法。论文的主要结果是数字 ${\sf crl}_p\left(M,g\right)$ resp。${\sf crl}\left(M,g\right)$ 仅在 $n$-sphere 上的圆形度量中达到其最大值 $2\pi$。