Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

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?)


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: http://dx.doi.org/10.1090/S0002-9947-04-03444-0
PII: S 0002-9947(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