Theorem of the three geodesics
In differential geometry the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every Riemannian manifold with the topology of a sphere has at least three simple closed geodesics (i.e. three embedded geodesic circles).[1] The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics of reversible Finsler 2-spheres. The theorem is sharp: although every Riemannian 2-sphere contains infinitely many distinct closed geodesics, only three of them are guaranteed to have no self-intersections. For example, by a result of Morse if the lengths of three principal axes of an ellipsoid are distinct, but sufficiently close to each other, then the ellipsoid has only three simple closed geodesics.[2]
History and proof
[edit]
A geodesic, on a Riemannian surface, is a curve that is locally straight at each of its points. On the Euclidean plane the geodesics are lines, and on a sphere the geodesics are great circles. The shortest path in the surface between two points is always a geodesic, but other geodesics may exist as well. A geodesic is said to be a closed geodesic if it returns to its starting point and starting direction; in doing so it may cross itself multiple times. The theorem of the three geodesics says that for surfaces homeomorphic to the sphere, there exist at least three non-self-crossing closed geodesics. There may be more than three; for instance, a sphere has infinitely many.
This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an ellipsoid, and from the study of geodesics on an ellipsoid, the shortest paths for ships to travel. In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators.[3] In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics,[4] and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture; while the general topological argument of the proof was correct, it employed a deformation result that was later found to be flawed.[5] Several authors proposed unsatisfactory solutions of the gap. A universally accepted solution was provided in the 1980s by Grayson, by means of the curve shortening flow[6]
Generalizations
[edit]A strengthened version of the theorem states that, on any Riemannian surface that is topologically a sphere, there necessarily exist three simple closed geodesics whose length is at most proportional to the diameter of the surface.[7]
The number of closed geodesics of length at most L on a smooth topological sphere grows in proportion to L/log L, but not all such geodesics can be guaranteed to be simple.[8]
On compact hyperbolic Riemann surfaces, there are infinitely many simple closed geodesics, but only finitely many with a given length bound. They are encoded analytically by the Selberg zeta function. The growth rate of the number of simple closed geodesics, as a function of their length, was investigated by Maryam Mirzakhani.[9]
The existence of three simple closed geodesics also holds for any reversible Finsler metric on the 2-sphere.[10]
Non-smooth metrics
[edit]It is also possible to define geodesics on some surfaces that are not smooth everywhere, such as convex polyhedra. The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses. Although some polyhedra have simple closed geodesics (for instance, the regular tetrahedron and disphenoids have infinitely many closed geodesics, all simple)[11][12] others do not. In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total angular defect of the vertices, and almost all polyhedra do not have such bisectors.[3][11]
Nevertheless, the theorem of the three geodesics can be extended to convex polyhedra by considering quasigeodesics, curves that are geodesic except at the vertices of the polyhedra and that have angles less than π on both sides at each vertex they cross. A version of the theorem of the three geodesics for convex polyhedra states that all polyhedra have at least three simple closed quasigeodesics; this can be proved by approximating the polyhedron by a smooth surface and applying the theorem of the three geodesics to this surface.[13] It is an open problem whether any of these quasigeodesics can be constructed in polynomial time.[14][15]
References
[edit]- ^ Poincaré, H. (1905), "Sur les lignes géodésiques des surfaces convexes" [Geodesics lines on convex surfaces], Transactions of the American Mathematical Society (in French), 6 (3): 237–274, doi:10.2307/1986219, JSTOR 1986219.
- ^ Ballmann, W.: On the lengths of closed geodesics on convex surfaces. Invent. Math. 71, 593–597 (1983)
- ^ a b Galperin, G. (2003), "Convex polyhedra without simple closed geodesics" (PDF), Regular & Chaotic Dynamics, 8 (1): 45–58, Bibcode:2003RCD.....8...45G, doi:10.1070/RD2003v008n01ABEH000231, MR 1963967.
- ^ Poincaré, H. (1905), "Sur les lignes géodésiques des surfaces convexes" [Geodesics lines on convex surfaces], Transactions of the American Mathematical Society (in French), 6 (3): 237–274, doi:10.2307/1986219, JSTOR 1986219.
- ^ Lyusternik, L.; Schnirelmann, L. (1929), "Sur le problème de trois géodésiques fermées sur les surfaces de genre 0" [The problem of three closed geodesics on surfaces of genus 0], Comptes Rendus de l'Académie des Sciences de Paris (in French), 189: 269–271.
- ^ Grayson, Matthew A. (1989), "Shortening embedded curves" (PDF), Annals of Mathematics, Second Series, 129 (1): 71–111, doi:10.2307/1971486, JSTOR 1971486, MR 0979601.
- ^ Liokumovich, Yevgeny; Nabutovsky, Alexander; Rotman, Regina (2017), "Lengths of three simple periodic geodesics on a Riemannian 2-sphere", Mathematische Annalen, 367 (1–2): 831–855, arXiv:1410.8456, Bibcode:2014arXiv1410.8456L, doi:10.1007/s00208-016-1402-5.
- ^ Hingston, Nancy (1993), "On the growth of the number of closed geodesics on the two-sphere", International Mathematics Research Notices, 1993 (9): 253–262, doi:10.1155/S1073792893000285, MR 1240637.
- ^ Mirzakhani, Maryam (2008), "Growth of the number of simple closed geodesics on hyperbolic surfaces", Annals of Mathematics, 168 (1): 97–125, doi:10.4007/annals.2008.168.97, MR 2415399, Zbl 1177.37036,
- ^ De Philippis, Guido; Marini, Michele; Mazzucchelli, Marco; Suhr, Stefan (2022), "Closed geodesics on reversible Finsler 2-spheres", Journal of Fixed Point Theory and Applications, 24 (2), arXiv:2002.00415, doi:10.1007/s11784-022-00962-9.
- ^ a b Fuchs, Dmitry [in German]; Fuchs, Ekaterina (2007), "Closed geodesics on regular polyhedra" (PDF), Moscow Mathematical Journal, 7 (2): 265–279, 350, doi:10.17323/1609-4514-2007-7-2-265-279, MR 2337883.
- ^ Cotton, Andrew; Freeman, David; Gnepp, Andrei; Ng, Ting; Spivack, John; Yoder, Cara (2005), "The isoperimetric problem on some singular surfaces", Journal of the Australian Mathematical Society, 78 (2): 167–197, doi:10.1017/S1446788700008016, MR 2141875.
- ^ Pogorelov, A. V. (1949), "Quasi-geodesic lines on a convex surface", Matematicheskii Sbornik, N.S., 25 (67): 275–306, MR 0031767.
- ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "24 Geodesics: Lyusternik–Schnirelmann", Geometric folding algorithms: Linkages, origami, polyhedra, Cambridge: Cambridge University Press, pp. 372–375, doi:10.1017/CBO9780511735172, ISBN 978-0-521-71522-5, MR 2354878.
- ^ Itoh, Jin-ichi; O'Rourke, Joseph; Vîlcu, Costin (2010), "Star unfolding convex polyhedra via quasigeodesic loops", Discrete and Computational Geometry, 44 (1): 35–54, arXiv:0707.4258, doi:10.1007/s00454-009-9223-x, MR 2639817.