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]
A geodesic, on a Riemannian surface, is a curve that is locally straight at each of its points. For instance, on the Euclidean plane the geodesics are lines, and on the surface of 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, the sphere itself 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 the 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
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 hyperbolicRiemann 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
Unsolved problem in computer science:
Is there an algorithm that can find a simple closed quasigeodesic on a convex polyhedron in polynomial time?
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
^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)
^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), doi:10.1007/s11784-022-00962-9.
^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.
theorem, three, geodesics, differential, geometry, theorem, three, geodesics, also, known, lyusternik, schnirelmann, theorem, states, that, every, riemannian, manifold, with, topology, sphere, least, three, simple, closed, geodesics, three, embedded, geodesic,. 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 Contents 1 History and proof 2 Generalizations 3 Non smooth metrics 4 ReferencesHistory and proof Edit A triaxial ellipsoid and its three geodesics A geodesic on a Riemannian surface is a curve that is locally straight at each of its points For instance on the Euclidean plane the geodesics are lines and on the surface of 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 the sphere itself 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 the 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 Poincare 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 EditA 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 EditUnsolved problem in computer science Is there an algorithm that can find a simple closed quasigeodesic on a convex polyhedron in polynomial time more unsolved problems in computer science 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 p 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 Poincare H 1905 Sur les lignes geodesiques 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 amp Chaotic Dynamics 8 1 45 58 Bibcode 2003RCD 8 45G doi 10 1070 RD2003v008n01ABEH000231 MR 1963967 Poincare H 1905 Sur les lignes geodesiques 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 probleme de trois geodesiques fermees sur les surfaces de genre 0 The problem of three closed geodesics on surfaces of genus 0 Comptes Rendus de l Academie 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 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 Vilcu 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 Retrieved from https en wikipedia org w index php title Theorem of the three geodesics amp oldid 1121530193, wikipedia, wiki, book, books, library,
