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

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 $P\subset M\setminus\{x,y\}$ such that every geodesic segment from $x$ to $y$ passes through a point of $P$. We show that for every closed $C^{\infty}$ manifold $M$ of dimension at least two and every pair $(x,y)\in M\times M$, there exists a dense $G_{\delta}$ set, $\mathcal{G}$, of $C^{\infty}$ Riemannian metrics on $M$ such that $(x,y)$ fails to have the finite blocking property for every $g\in\mathcal{G}$. Moreover, there exists a dense $G_{\delta}$ set, $\mathcal{G}_{1}$, of $C^{\infty}$ Riemannian metrics on $M$ such that for every $g\in\mathcal{G}_{1}$, there is a dense $G_{\delta}$ subset $\mathcal{R=R}(g)$ of $M\times M$ such that every $(x,y)\in\mathcal{R}$ fails to have the finite blocking property for $g$.

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Published 20 May 2011