2010/05/27 by Vladimir Chernov, Paul Kinlaw, Rustam Sadykov
Computer Science · Mathematics · Physics and Astronomy · #Cohomology #Combinatorics #Geodesic #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Riemannian manifold #Topological and Geometric Data Analysis #Topology (electrical circuits) #gr-qc #math-ph #math.DG #math.GT #math.MP #msc:53C20 #msc:53C22 #msc:53C50 #msc:57R17
paper · pdf · doi:10.1016/j.geomphys.2010.05.010
published as J.Geom.Phys.60:1530-1538,2010 · 14 pages
arxiv created 2010/05/27 · openalex publication_date 2010/05/28 · arxiv updated 2014/11/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A complete Riemannian manifold (M, g) is a Yxl-manifold if every unit speed geodesic γ(t) originating at γ(0)=x∈ M satisfies γ(l)=x for 0≠ l∈ \R. Bérard-Bergery proved that if (Mm,g), m>1 is a Yxl-manifold, then M is a closed manifold with finite fundamental group, and the cohomology ring H^*(M, \Q) is generated by one element. We say that (M,g) is a Yx-manifold if for every ε>0 there exists l>ε such that for every unit speed geodesic γ(t) originating at x, the point γ(l) is ε-close to x. We use Low's notion of refocussing Lorentzian space-times to show that if (Mm, g), m>1 is a Yx-manifold, then M is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a Yx-manifold is a Yx-manifold. Another corollary is that if (Mm,g), m=2,3 is a Yx-manifold, then (M, h) is a Yxl-manifold for some metric h.