Introduction to systolic geometry

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.

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 familiar shapes of drops of water express minima of surface area.

In geometry, a systole is a distance which is characteristic of a compact metric space which is not simply connected.

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.

Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water.

The solution to the isoperimetric problem in the plane is usually expressed in the form of an inequality that relates the length

The isoperimetric inequality states that and that the equality holds if and only if the curve is a round circle.

The isoperimetric inequality is an upper bound for area in terms of length.

There is a geometric inequality which provides an upper bound for a certain length in terms of area.

It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.

admits a pair of opposite (antipodal) points and a path of length

We will denote it as follows: Note that a loop minimizing length is necessarily a closed geodesic.

is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by William Tutte.

Possibly inspired by Tutte's article, Charles Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student P. M. Pu.

The actual term systole itself was not coined until a quarter century later, by Marcel Berger.

This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter.

Referring to these systolic inequalities, Thom reportedly exclaimed: Mais c'est fondamental!

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the March 2008 issue of the Notices of the American Mathematical Society.

Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals.

This distance function corresponds to the metric of constant Gaussian curvature +1.

can be defined as the surface obtained by identifying each pair of antipodal points on the 2-sphere.

Pu's inequality for the real projective plane applies to general Riemannian metrics on

A student of Charles Loewner's, Pao Ming Pu proved in a 1950 thesis (published in 1952) that every metric

The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature.

To state his result, one requires a topological notion of an essential manifold.

is the area of the region bounded by a closed Jordan curve of length (perimeter)

The explanation of the strengthened version of Loewner's inequality is somewhat more technical than the rest of this article.

The strengthened version is the inequality where Var is the probabilistic variance while f is the conformal factor expressing the metric g in terms of the flat metric of unit area in the conformal class of g. The proof results from a combination of the computational formula for the variance and Fubini's theorem (see Horowitz et al, 2009).