Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-04-03444-0
Published electronically: January 23, 2004
MathSciNet review: 2048529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $M$.


References [Enhancements On Off] (What's this?)

  • [1917] G. D. Birkhoff, Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18, 199-300 (1917).
  • [1930] W. Blaschke, Vorlesungen über Differentialgeometrie I, 3. Aufl. Julius Springer, Berlin (1930). MR 17:888g
  • [1983] R. Bott, Lectures on Morse Theory, Old and New. Proc. Sympos. Pure Math. vol. 39, part 2, 3-30 (1983). MR 84m:58026b
  • [1982] C. Croke, Poincaré's Problem and the Length of the Shortest Closed Geodesic on a Convex Hypersurface. J. Differential Geom. 17, 595-634 (1982). MR 84f:58034
  • [2002] W. P. A. Klingenberg, Klassische Differentialgeometrie. ftp://www.math.uni-bonn.de/ 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] S. Smale, Regular curves on Riemannian manifolds. Trans. Amer. Math. Soc. 87, 2492-502 (1958). MR 20:1319
  • [1992] I. A. Ta{\u{\i}}\kern.15emmanov, On the existence of three nonselfintersecting closed geodesics on manifolds homeomorphic to the 2-sphere. Russian Math. (Iz. VUZ) 40, 565-590 (1993). (Translation of a paper which appeared 1992). MR 93k:58051

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A05, 53C22, 34C25, 58G30, 58E10

Retrieve articles in all journals with MSC (2000): 53A05, 53C22, 34C25, 58G30, 58E10


Additional Information

Wilhelm P. A. Klingenberg
Affiliation: Mathematisches Institut der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany
Email: klingenb@math.uni-bonn.de

DOI: https://doi.org/10.1090/S0002-9947-04-03444-0
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

American Mathematical Society