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 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.
| 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 |
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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
Homology classes of the circle space on spheres and the discontinuity of deformations. / Zhou, Congyi; Long, Yiming.
In: Advanced Nonlinear Studies, Vol. 5, No. 1, 01.01.2005, p. 1-11.Research output: Contribution to journal › Article
}
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 -