Robert's Mistakes

Systolic geometry

Apr 16, 2020

A systole is the shortest loop you can put in a space that cannot be contracted further. Systolic geometry is about estimating properties of that space given the length of its systole.

Consider a torus. It doesn't have to have the same thickness everywhere. Now, the systole will be a loop at the thinnest spot. Using its length we can give a lower bound for the surface of the torus. Think about it, when you shrink the torus too much you'll find a better place to put a shorter systole.

The length of the systole gives a lower bound for the area of a torus! This idea can be generalized...

It's only a little weird to relate a length to an area. At least, there are quite a few classic theorems relating length and area. But in higher dimensions the number of known theorems relating length and n-volume quickly thin out.

Here's a cool intro about #systolic #geometry: Larry Guth - Metaphors in Systolic Geometry


Here's a suggestive bit from Wikipedia:

Any centrally symmetric convex body of surface area A can be squeezed through a noose of length sqrt(π A), with the tightest fit achieved by a sphere. This property is equivalent to a special case of Pu's inequality, one of the earliest systolic inequalities.

Pu's inequality is a special case in systolic geometry telling something about the total area of the projective plane, given the length of its systole.

Systolic geometry is special because it provides theorems without any constraints on the metric! To measure the length of a curve we need metric, but with increasing dimension the ways to define a metric escalate, and very weird ones appear.