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

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.