A dense G-delta set of Riemannian metrics without the finite blocking property

TL;DR

This paper proves that on any closed smooth manifold of dimension at least two, there exists a dense G-delta set of Riemannian metrics for which no pair of points has the finite blocking property, highlighting the generic failure of this property.

Contribution

It establishes the existence of a dense G-delta set of metrics on any closed manifold where the finite blocking property fails for all point pairs, extending understanding of geodesic behavior.

Findings

Finite blocking property fails generically for dense G-delta sets of metrics.

For every point pair, there exists a metric where the property does not hold.

The result applies to all closed manifolds of dimension at least two.

Abstract

A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) in M x M, there exists a dense G-delta set of C-infinity Riemannian metrics on M such that (x,y) fails to have the finite blocking property for every g in that set.