×

Simple closed geodesics on most Alexandrov surfaces. (English) Zbl 1317.53094

The authors establish various generic results regarding the existence of simple closed geodesics on the Baire space of Alexandrov surfaces. These results point out the dependence on the curvature bound and the topology of the surface. They cover different cases of typical surfaces.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53C22 Geodesics in global differential geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adiprasito, K.; Zamfirescu, T., Few Alexandrov spaces are Riemannian, J. Nonlinear Convex Anal. (2015), in press · Zbl 1368.53047
[2] Alexandrov, A. D., Die innere Geometrie der konvexen Flächen (1955), Akademie Verlag: Akademie Verlag Berlin · Zbl 0065.15102
[3] Angenent, S., Curve shortening and the topology of closed geodesics on surfaces, Ann. of Math., 162, 1185-1239 (2005)
[4] Bangert, V., On the existence of closed geodesics on two-spheres, Int. J. Math., 4, 1-10 (1993) · Zbl 0791.53048
[5] Bezdek, K., Ball-polyhedra as intersections of congruent balls, (Classical Topics in Discrete Geometry. Classical Topics in Discrete Geometry, CMS Books Math. (2010), Springer: Springer New York), 57-68
[6] Burago, D.; Burago, Yu.; Ivanov, S., A Course in Metric Geometry (2001), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0981.51016
[7] Burago, Yu.; Gromov, M.; Perel’man, G., A.D. Alexandrov spaces with curvature bounded below, Russian Math. Surveys, 47, 1-58 (1992), (in English; Russian original) · Zbl 0802.53018
[8] Contreras, G., Geodesic flows with positive topological entropy, twist maps and hyperbolicity, Ann. of Math., 172, 761-808 (2010) · Zbl 1204.37032
[9] Franks, J., Geodesics on \(S^2\) and periodic points of annulus homeomorphisms, Invent. Math., 108, 403-418 (1992) · Zbl 0766.53037
[10] Gromov, M.; Lafontaine, J.; Pansu, P., Structure métrique pour les variétés riemanniennes (1981), CEDIC/Fernand Nathan
[11] Gruber, P., Geodesics on typical convex surfaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 82, 651-659 (1988) · Zbl 0741.52006
[12] Gruber, P., A typical convex surface contains no closed geodesic, J. Reine Angew. Math., 416, 195-205 (1991) · Zbl 0718.52003
[13] Gruber, P., Baire categories in convexity, (Gruber, P.; Wills, J., Handbook of Convex Geometry, vol. B (1993), North-Holland: North-Holland Amsterdam), 1327-1346 · Zbl 0791.52002
[14] Hadamard, J., Les surfaces à courbures opposées et leurs lignes géodésique, J. Math. Pures Appl., 4, 27-75 (1898) · JFM 29.0522.01
[15] Itoh, J.; Rouyer, J.; Vîlcu, C., Moderate smoothness of most Alexandrov surfaces, Int. J. Math. (2015), in press · Zbl 1325.53093
[16] Kapovitch, V., Perelman’s stability theorem, (Cheeger, J.; etal., Metric and Comparison Geometry. Metric and Comparison Geometry, Surv. Differ. Geom., vol. 11 (2007), International Press), 103-136 · Zbl 1151.53038
[17] Klingenberg, W., Riemannian Geometry (1982), De Gruyter: De Gruyter Berlin, New York · Zbl 0495.53036
[18] Mirzakhani, M., Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math., 168, 97-125 (2008) · Zbl 1177.37036
[20] Pogorelov, A. V., Quasigeodesic lines on convex surfaces, Mat. Sb., 25, 275-307 (1949)
[21] Pogorelov, A. V., Extrinsic Geometry of Convex Surfaces (1973), Amer. Math. Soc. · Zbl 0311.53067
[22] Rademacher, H., On a generic property of geodesic flows, Math. Ann., 298, 101-116 (1994) · Zbl 0813.53029
[23] Rouyer, J., Generic properties of compact metric spaces, Topology Appl., 158, 2140-2147 (2011) · Zbl 1237.54025
[24] Rouyer, J.; Vîlcu, C., The connected components of the space of Alexandrov surfaces, (Ibadula, D.; Veys, W., Bridging Algebra, Geometry and Topology. Bridging Algebra, Geometry and Topology, Springer Proc. Math. Stat., vol. 96 (2014)), 249-254 · Zbl 1338.53095
[25] Rouyer, J.; Vîlcu, C., Farthest points on most Alexandrov surfaces · Zbl 1435.53052
[26] Shiohama, K., An Introduction to the Geometry of Alexandrov Spaces, Lect. Notes Ser. (1992), Seoul National University · Zbl 0826.53001
[27] Toponogov, V. A., Computation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR, 124, 282-284 (1959), (in Russian) · Zbl 0092.14603
[28] Troyanov, M., Metrics of constant curvature on a sphere with two conical singularities, (Third International Symposium on Differential Geometry at Peñiscola. Third International Symposium on Differential Geometry at Peñiscola, 1988, Peñiscola. Third International Symposium on Differential Geometry at Peñiscola. Third International Symposium on Differential Geometry at Peñiscola, 1988, Peñiscola, Lecture Notes in Math., vol. 1410 (1989), Springer: Springer Berlin), 296-306
[29] Zamfirescu, T., Baire categories in convexity, Atti Semin. Mat. Fis. Univ. Modena, 39, 139-164 (1991) · Zbl 0780.52003
[30] Zamfirescu, T., Long geodesics on convex surfaces, Math. Ann., 293, 109-114 (1992) · Zbl 0735.53032
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.