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

Congyi Zhou, Yiming Long

    Research output: Contribution to journalArticle

    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 - Jan 1 2005

    Fingerprint

    homology
    Homology
    Discontinuity
    discontinuity
    Circle
    theorems
    Geodesies
    Riemannian geometry
    Self-intersection
    polygons
    Theorem
    Proposition
    intersections
    closures
    Polygon
    Geodesic
    Counterexample
    Genus
    Closure
    counters

    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 journalArticle

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