Topological properties of manifolds admitting a Y^{x}Riemannian metric
Abstract
A complete Riemannian manifold (M,g) is a Ylxmanifold if every unit speed geodesic γ(t) originating at γ(0)=x∈M satisfies γ(l)=x for 0≠l∈R. BérardBergery proved that if (M^{m},g),m>1 is a Ylxmanifold, 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 Y^{x}manifold if for every ∊>0 there exists l>∊ such that for every unit speed geodesic γ(t) originating at x, the point γ(l) is ∊close to x. We use Low's notion of refocussing Lorentzian spacetimes to show that if (M^{m},g),m>1 is a Y^{x}manifold, then M is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a Y^{x}manifold is a Y^{x}manifold. Another corollary is that if (M^{m},g),m=2,3 is a Y^{x}manifold, then (M,h) is a Ylxmanifold for some metric h.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 October 2010
 DOI:
 10.1016/j.geomphys.2010.05.010
 arXiv:
 arXiv:1005.5075
 Bibcode:
 2010JGP....60.1530C
 Keywords:

 Mathematics  Differential Geometry;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 Mathematics  Geometric Topology;
 Primary 53C20;
 Secondary 53C22;
 53C50;
 57R17
 EPrint:
 14 pages