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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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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

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