The length of a shortest closed geodesic on a two-dimensional sphere and coverings by metric balls.

*(English)*Zbl 1080.53031Summary: In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold \(M\) diffeomorphic to the standard two-dimensional sphere. The first result is that the length \(l(M)\) of a shortest closed geodesic is bounded from above by \(4r\), where \(r\) is the radius of \(M\). (In particular this means that \(l(M)\) is bounded from above by \(2d\), if \(M\) can be covered by a ball of radius \(d/2\), where \(d\) is the diameter of \(M\).) The second result is that \(l(M)\) is bounded from above by \(2(\max\{r_1,r_2\} + r_1 + r_2)\), if \(M\) can be covered by two closed metric balls of radii \(r_{1}, r_{2}\) respectively. For example, if \(r_{1} = r_{2} = d/2 \), then \(l(M) \leq 3d\). The third result is that \(l(M) \leq 2 (\max\{r_1,r_2,r_3\} + r_1 + r_2 + r_3)\), if \(M\) can be covered by three closed metric balls of radii \(r_{1}, r_{2}, r_{3}\). Finally, we present an estimate for \(l(M)\) in terms of radii of \(k\) metric balls covering \(M\), where \(k \geq 3\), if these balls have a special configuration.

##### MSC:

53C22 | Geodesics in global differential geometry |

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