Secure billiards and Serre’s thesis

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 A,B such that no finite set of points can block all billiard trajectories from A to B

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 M to be secure if for any pair of points A, B there exist a finite set of points S\subset M-\{A,B\} such that all geodesics from A to B go through a point of S.

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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