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

Congyi Zhou, Yiming Long

    Research output: Contribution to journalArticlepeer-review

    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 languageEnglish (US)
    Pages (from-to)1-11
    Number of pages11
    JournalAdvanced Nonlinear Studies
    Volume5
    Issue number1
    DOIs
    StatePublished - Feb 2005

    Keywords

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

    ASJC Scopus subject areas

    • Statistical and Nonlinear Physics
    • Mathematics(all)

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