## Linear bounds for lengths of geodesic loops on Riemannian 2-spheres.(English)Zbl 1243.53075

Summary: Let $$M$$ be a closed surface diffeomorphic to $$S^2$$ endowed with a Riemannian metric. Denote the diameter of $$M$$ by $$d$$. We prove that for every $$x \in M$$ and every positive integer $$k$$ there exist $$k$$ distinct geodesic loops based at $$x$$ of length $$\leq 20kd$$.

### MSC:

 53C22 Geodesics in global differential geometry

### Keywords:

diameter; geodesic loop; length
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