×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bott, R., Lectures on Morse theory, old and new, Bull. Amer. Math. Soc., 7, 331-358, (1982) · Zbl 0505.58001
[2] Croke, C. B., Area and the length of the shortest closed geodesic, J. Differential Geom., 27, 1-21, (1988) · Zbl 0642.53045
[3] Maeda, M., The length of a closed geodesic on a compact surface, Kyushu J. Math, 48, 9-18, (1994) · Zbl 0818.53064
[4] Nabutovsky, A. and Rotman, R.: The length of a shortest closed geodesic on a two-dimensional sphere, IMRN No. 23, 1211-1222. · Zbl 1003.53030
[5] Sabourau, S.: Filling radius and short closed geodesics of the sphere, Preprint. · Zbl 1064.53020
[6] Calabi, E.; Cao, J., Simple closed geodesics on convex surfaces, J. Differential Geom., 36, 517-549, (1992) · Zbl 0768.53019
[7] Almgren, F., The homotopy groups of the integral cycle groups, Topology, 1, 257-299, (1962) · Zbl 0118.18503
[8] Pitts, J.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Math Notes 27, Princeton Univ. Press, Princeton, NJ and Univ. Tokyo Press, Tokyo, 1981. · Zbl 0462.58003
[9] Nabutovsky, A. and Rotman, R.: Volume, diameter and the minimal mass of a stationary 1-cycle, preprint, arXiv:math.DG/\(0201269 v^2\).Available at http://arXiv.org/abs/math. DG/0201269.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.