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The length of a shortest closed geodesic on a two-dimensional sphere and coverings by metric balls. (English) Zbl 1080.53031
Summary: 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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