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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Poincaré's closed geodesic on a convex surface

Author: Wilhelm P. A. Klingenberg
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2545-2556
MSC (2000): Primary 53A05, 53C22; Secondary 34C25, 58G30, 58E10
Published electronically: January 23, 2004
MathSciNet review: 2048529
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Abstract: We present a new proof for the existence of a simple closed geodesic on a convex surface $M$. This result is due originally to Poincaré. The proof uses the ${2k}$-dimensional Riemannian manifold ${_{k}\Lambda M} = \hbox {(briefly)} \Lambda $ of piecewise geodesic closed curves on $M$ with a fixed number $k$ of corners, $k$ chosen sufficiently large. In $\Lambda $ we consider a submanifold $\overset{\approx }{\Lambda }_{0}$ formed by those elements of $\Lambda $ which are simple regular and divide $M$ into two parts of equal total curvature $2\pi $. The main burden of the proof is to show that the energy integral $E$, restricted to $\overset{\approx }{\Lambda }_{0}$, assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on $M$.

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  • [1917] G. D. Birkhoff, Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18, 199-300 (1917).
  • [1930] Wilhelm Blaschke, Vorlesungen über Integralgeometrie, Deutscher Verlag der Wissenschaften, Berlin, 1955 (German). 3te Aufl. MR 0076373
  • [1983] Raoul Bott, Lectures on Morse theory, old and new, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Science Press, Beijing, 1982, pp. 169–218. MR 714336
  • [1982] Christopher B. Croke, Poincaré’s problem and the length of the shortest closed geodesic on a convex hypersurface, J. Differential Geom. 17 (1982), no. 4, 595–634 (1983). MR 683167
  • [2002] W. P. A. Klingenberg, Klassische Differentialgeometrie. people/klingenb/.
  • [1929] L. Lusternik and L. Schnirelmann, Sur le problème des trois géodésiques fermées sur les surfaces de genre $0$. C. R. Acad. Sci. Paris Sér. I Math. 189, 269-271 (1929).
  • [1905] H. Poincaré, Sur les lignes géodésiques sur les surfaces convexes. Trans. Amer. Math. Soc. 17, 237-274 (1909).
  • [1958] Stephen Smale, Regular curves on Riemannian manifolds, Trans. Amer. Math. Soc. 87 (1958), 492–512. MR 0094807, 10.1090/S0002-9947-1958-0094807-0
  • [1992] I. A. Taĭmanov, On the existence of three nonintersecting closed geodesics on manifolds that are homeomorphic to the two-dimensional sphere, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 3, 605–635 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 3, 565–590. MR 1188331, 10.1070/IM1993v040n03ABEH002177

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Additional Information

Wilhelm P. A. Klingenberg
Affiliation: Mathematisches Institut der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany

Received by editor(s): April 16, 2003
Received by editor(s) in revised form: June 3, 2003
Published electronically: January 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society