Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


IMPACT FACTOR 2018: 1.650

CiteScore 2018: 1.49

SCImago Journal Rank (SJR) 2018: 1.422
Source Normalized Impact per Paper (SNIP) 2018: 0.865

Mathematical Citation Quotient (MCQ) 2018: 1.19

Online
ISSN
2169-0375
See all formats and pricing
More options …
Volume 5, Issue 1

Issues

Homology Classes of the Circle Space on Spheres and the Discontinuity of Deformations

Congyi Zhou / Yiming Longy
Published Online: 2016-03-10 | DOI: https://doi.org/10.1515/ans-2005-0101

Abstract

The famous Lusternik-Schnirelmann theorem claims that on a surface of genus 0 there exist at least three simple closed geodesics 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 con- structions 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.

Keywords: Lusternik-Schnirelmann theorem; 2-sphere; non-trivial homology class; circle space; deformation; discontinuity

About the article

Received: 2004-02-27

Published Online: 2016-03-10

Published in Print: 2005-02-01


Citation Information: Advanced Nonlinear Studies, Volume 5, Issue 1, Pages 1–11, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2005-0101.

Export Citation

© 2016 by Advanced Nonlinear Studies, Inc..

Comments (0)

Please log in or register to comment.
Log in