A few days ago, Sergei Tabachnikov put a preprint on the arXiv called *Birkhoff billiards are insecure*. Now that’s a catchy title, so I had to at least browse the paper.

The thing is, it immediatly rang a bell, since the abstract says:

*We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points such that no finite set of points can block all billiard trajectories from to *

while I had read last year a nice history book by Marcel Berger which provides an overview of mathematics in France in the past five centuries (I recommend it to french high-school students), and where it is written (my translation):

*already with his thesis in 1951, Serre became famous by using the Leray spectral sequence to show […] that [for any compact Riemannian manifold] any pair of points can be joined by an infinity of geodesics* (Those lucky enough to have a JSTOR login can read Serre’s thesis here. Those who don’t can still view large excerpts on google books.)

Obviously there’s a striking similarity between the two. Tabashnikov defines a manifold to be secure if for any pair of points there exist a finite set of points such that all geodesics from to go through a point of .

So Serre’s result implies a statement similar to Tabashnikov’s abstract, but for compact riemannian manifolds: they are all insecure.

*EDIT*: JSTOR link corrected, added google books link.

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