## Of Loewner and Besicovitch

I’d like to share some of the notes that I’m writing up about systoles. After a little bit of preliminaries we’ll see a slick proof the systolic inequality in the torus case.

The systole of manifold is the length of the shortest non-contractible curve in the manifold. Systoles hard to estimate. In general there are many many non-contractible curves, and its not easy to track down which one should be smallest. If someone hands you a donut, you’ll visually guess the systole correctly. If someone hands you a coffee cup, it’s still clear. Once you get a generic metric, you’re in deep water. Loewner‘s theorem gives us an upper bound on the systole a Riemannian 2-torus (generalized donut / coffee cup case).

Definition 1Given a connected closed Riemannian -manifold thesystoleof is . That is, the systole is the length of the shortest non-contractible curve in .

Theorem 2If is a Riemannian 2-torus then:Moreover, equality occurs if and only if is the flat torus where is a third root of unity.

The equality case described above is worth exploring. Let denote the torus described in the equality case. We take the plane, here written as and we mod out by a particular lattice: . This lattice has a fundamental domain the parallelogram bounded by , and . We then get that . Thus, the upper bound in Loewner’s theorem in one. We certainly have many curves which realize this bound. Any curve parallel to the edge or the edge will do. This shows we now check that one is the shortest length of a non-contractible curve. We’ll look at things in , the universal cover of . Consider a curve starting from the origin of length less than one. If is non-contractible then it has to be closed and it can’t end at the origin, since otherwise we could contract it. The nearest points in that get identified with the origin are . One can see that to hit such a point, must have length at least one. Thus, there are no non-contractible curves of length less than one. Thus equality holds in Loewner’s inequality. Checking that the lattice described above maximizes the systole volume ratio is a good exercise in flat geometry.

We’re going to outline a proof that doesn’t give us the sharp constant, but which is quite instructive and natural. The reference for this is Gromov’s wonderful book: Metric Structures. First we introduce the following useful quantitative lemma:

Lemma 3 (Besicovitch)Let denote a closed unit interval and let be a Riemannian square. Let and denote the distances between pairs of opposite sides. One has:

Now we’ll deduce Theorem~2 (without the sharp constant) from the lemma above.

*Proof:*

First note that if we cut along the shortest non-seperating curve we obtain two boundary components . Let be the shortest curve homotopic to one of these boundary components. In particular, . We apply the co-area formula. We have . So, . This is quite nice. Pick an arc from to realizing . Note that its end points might not be equal and so we complete it to a curve by adding in an arc of of length at most . We then have .

We then apply Besicovitch and obtain:

Eventually I’ll get to writing about what this is useful for and how it’s been generalized. Loewner’s systolic inequality inspired a lot of good work. For much more inspired and in-depth introduction to systolic inequalities one can do no better than Larry Guth‘s article: Metaphors in systolic Geometry.

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