Real Analytic Metrics on S² with Total Absence of Finite Blocking

2011/09/07 by Marlies Gerber, Gerber, Marlies, Lihuei Liu +1 · 1 citation
Mathematics · Physics and Astronomy · #37D25 #37D40 #53C22 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DG #msc:37D25 #msc:37D40 #msc:53C22

paper · pdf · doi:10.48550/arxiv.1109.1336

30 pages, 5 figures

arxiv created 2011/09/07 · openalex publication_date 2011/09/07 · arxiv updated 2011/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If (M,g) is a Riemannian manifold and x,y are points in M, then a subset P of M\x,y is said to be a blocking set for (x,y) if every geodesic from x to y passes through a point of P. If no pair (x,y) in M X M has a finite blocking set, then (M,g) is said to be totally insecure. We prove that there exist real analytic metrics h on S² such that (S²,h) is totally insecure.

Cited by

• Connection blocking in homogeneous spaces and nilmanifolds

Related

• A dense G-delta set of Riemannian metrics without the finite blocking property
• Blocking: New examples and properties of products
• Secure two-dimensional tori are flat
• (In)equality distance patterns and embeddability into Hilbert spaces
• Connection blocking in homogeneous spaces and nilmanifolds