### Abstract

The famous Lusternik-Schnirelmann theorem claims that on a surface of genus 0 there exist at least three simple closed geodesies without self-intersections. A variational proof of this theorem is given in the book "Riemannian Geometry (2ed Ed.)" of W. Klingenberg. In this paper, firstly we point out that the construction of the three non-trivial relative homology classes in the circle space on S^{2} in this proof (the proof of Proposition 3.7.19 of [7]) is incorrect, and give explicit constructions of these homology classes. Secondly, we construct a counter-example to show that the deformation constructed in this proof in the closure of non-self- intersecting geodesic polygons (on pp.343-344 of [7]) is discontinuous, and therefore this proof is not complete.

Original language | English (US) |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Advanced Nonlinear Studies |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2005 |

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### Keywords

- 2-Sphere
- Circle space
- Deformation
- Discontinuity
- Lusternik-Schnirelmann theorem
- Non-trivial homology class

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)

### Cite this

*Advanced Nonlinear Studies*,

*5*(1), 1-11. https://doi.org/10.1515/ans-2005-0101