2022/03/14 by Herng Yi Cheng, Cheng, Herng Yi
Mathematics · Computer Science · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2203.07166
For each odd n ≥ 3, we construct a closed convex hypersurface of ℝn+1 that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even n, so in a sense we prove that Synge's theorem is "sharp." We also construct stable figure-eights: that is, for each n ≥ 3 we embed the figure-eight graph in a closed convex hypersurface of ℝn+1, such that sufficiently small variations of the embedding either preserve its image or must increase its length. These index-zero geodesics and stable figure-eights are mainly derived by constructing explicit billiard trajectories with "controlled parallel transport" in convex polytopes.