Introduction to systolic geometry

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Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.

Mikhail Gromov once voiced the opinion that the isoperimetric inequality was known already to the Ancient Greeks. The mythological tale of Dido, Queen of Carthage shows that problems about making a maximum area for a given perimeter were posed in a natural way, in past eras.

The relation between length and area is closely related to the physical phenomenon known as surface tension, which gives a visible form to the comparable relation between surface area and volume. The familiar shapes of drops of water express minima of surface area.

The purpose of this article is to explain another such relation between length and area. A space is called simply connected if every loop in the space can be contracted to a point in a continuous fashion. For example, a room with a pillar in the middle, connecting floor to ceiling, is not simply connected. In geometry, a systole is a distance which is characteristic of a compact metric space which is not simply connected. It is the length of a shortest loop in the space that cannot be contracted to a point in the space. In the room example, absent other features, the systole would be the circumference of the pillar. Systolic geometry gives lower bounds for various attributes of the space in terms of its systole.

It is known that the Fubini–Study metric is the natural metric for the geometrisation of quantum mechanics. In an intriguing connection to global geometric phenomena, it turns out that the Fubini–Study metric can be characterized as the boundary case of equality in Gromov's inequality for complex projective space, involving an area quantity called the 2-systole, pointing to a possible connection to quantum mechanical phenomena.

In the following, these systolic inequalities will be compared to the classical isoperimetric inequalities, which can in turn be motivated by physical phenomena observed in the behavior of a water drop.