### 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) |
---|---|

Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Advanced Nonlinear Studies |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2005 |

### Fingerprint

### 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

**Homology classes of the circle space on spheres and the discontinuity of deformations.** / Zhou, Congyi; Long, Yiming.

Research output: Contribution to journal › Article

*Advanced Nonlinear Studies*, vol. 5, no. 1, pp. 1-11. https://doi.org/10.1515/ans-2005-0101

}

TY - JOUR

T1 - Homology classes of the circle space on spheres and the discontinuity of deformations

AU - Zhou, Congyi

AU - Long, Yiming

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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 S2 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.

AB - 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 S2 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.

KW - 2-Sphere

KW - Circle space

KW - Deformation

KW - Discontinuity

KW - Lusternik-Schnirelmann theorem

KW - Non-trivial homology class

UR - http://www.scopus.com/inward/record.url?scp=16244422672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=16244422672&partnerID=8YFLogxK

U2 - 10.1515/ans-2005-0101

DO - 10.1515/ans-2005-0101

M3 - Article

AN - SCOPUS:16244422672

VL - 5

SP - 1

EP - 11

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

SN - 1536-1365

IS - 1

ER -