Introduction to systolic geometry

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This article is intended as a generally accessible introduction to the subject. For the main encyclopedia article, please see Systolic geometry.

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[edit] Surface tension and shape of a water drop

Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere. Thus the round shape of the drop is a consequence of the phenomenon of surface tension. Mathematically, this phenomenon is expressed by the isoperimetric inequality.

[edit] Isoperimetric inequality in the plane

The solution to the isoperimetric problem in the plane is usually expressed in the form of an inequality that relates the length L of a closed curve and the area A of the planar region that it encloses. The isoperimetric inequality states that

4\pi A \le L^2,\,

and that the equality holds if and only if the curve is a round circle. The inequality is an upper bound for area in terms of length. It can be rewritten as follows:

L^2 -4\pi A \geq 0. \,

[edit] Central symmetry

Recall the notion of central symmetry: a Euclidean polyhedron is called centrally symmetric if it is invariant under the "antipodal" map

x \mapsto -x. \,

Thus, in the plane central symmetry is the rotation by 180 degrees. For example, an ellipse is centrally symmetric, as is any ellipsoid in 3-space.

[edit] Property of a centrally symmetric polyhedron in 3-space

There is a geometric inequality that is in a sense dual to the isoperimetric inequality in the following sense. Both involve a length and an area. 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. More precisely it can be described as follows.

Any centrally symmetric convex body of surface area A can be squeezed through a noose of length \sqrt{\pi 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.

For example, an ellipsoid is an example of a convex centrally symmetric body in 3-space. It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.

An alternative formulation is as follows. Every convex centrally symmetric body P in {\mathbb R}^3 admits a pair of opposite (antipodal) points and a path of length L joining them and lying on the boundary \partial P of P, satisfying

L^2 \leq \frac{\pi}{4} \mathrm{area}(\partial P).

[edit] Notion of systole

The systole of a compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X. We will denote it as follows:

\mathrm{sys}(X). \,

When X 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! [These results are of fundamental importance!]

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the march '08 issue of the Notices of the American Mathematical Society. A bibliography at the Website for systolic geometry and topology currently contains over 170 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link has emerged with the Lusternik-Schnirelmann category. The existence of such a link can be thought of as a theorem in systolic topology.

[edit] The real projective plane

In projective geometry, the real projective plane \mathbb {RP}^2 is defined as the collection of lines through the origin in \mathbb {R}^3. The distance function on \mathbb {RP}^2 is most readily understood from this point of view. Namely, the distance between two lines through the origin is by definition the angle between them (measured in radians), or more precisely the lesser of the two angles. This distance function corresponds to the metric of constant Gaussian curvature +1.

Alternatively, \mathbb {RP}^2 can be defined as the surface obtained by identifying each pair of antipodal points on the 2-sphere.

Other metrics on \mathbb {RP}^2 can be obtained by quotienting metrics on S2 imbedded in 3-space in a centrally symmetric way.

Topologically, \mathbb {RP}^2 can be obtained from the Mobius strip by attaching a disk along the boundary.

Among closed surfaces, the real projective plane is the simplest non-orientable such surface.

[edit] Pu's inequality

Pu's inequality applies to general Riemannian metrics on \mathbb {RP}^2.

A student of Charles Loewner's, P.M. Pu proved in a 1950 thesis (published in 1952) that every metric g on the real projective plane \mathbb {RP}^2 satisfies the optimal inequality

\mathrm{sys}(g)^2 \leq \frac{\pi}{2} \mathrm{area}(g),

where sys is the systole. The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature. Alternatively, the inequality can be presented as follows:

\mathrm{area}(g) - \frac{2}{\pi} \mathrm{sys}(g)^2 \geq 0.

There is a vast generalisation of Pu's inequality, due to Mikhail Gromov, called Gromov's systolic inequality for essential manifolds. To state his result, one requires a topological notion of an essential manifold.

[edit] Loewner's torus inequality

Similarly to Pu's inequality, Loewner's torus inequality relates the total area, to the systole, i.e. least length of a noncontractible loop on the torus (T2,g):

\mathrm{area}(g) - \tfrac{\sqrt{3}}{2} \mathrm{sys}(g)^2 \geq 0.

The boundary case of equality is attained if and only if the metric is homothetic to the flat metric obtained as the quotient of {\mathbb R}^2 by the lattice formed by the Eisenstein integers.

[edit] Bonnesen's inequality

The classical Bonnesen's inequality is the strengthened isoperimetric inequality

L^2 - 4\pi A \geq \pi^2(R-r)^2. \,

Here A is the area of the region bounded by a closed Jordan curve of length (perimeter) L in the plane, R is the circumradius of the bounded region, and r is its inradius. The error term π2(R − r)2 on the right hand side is traditionally called the isoperimetric defect. There exists a similar strengthening of Loewner's inequality.

[edit] References

  • Bangert, V.; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture and ovalless real hyperelliptic surfaces. Geometric and Functional Analysis (GAFA) 15 (2005), no. 3, 577-597.
  • Berger, M.: Systoles et applications selon Gromov. (French. French summary) [Systoles and their applications according to Gromov] Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 771, 5, 279--310.
  • Berger, M.: A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003.
  • Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374-376.
  • Buser, P.; Sarnak, P.: On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117 (1994), no. 1, 27--56.
  • Gromov, M. Systoles and intersystolic inequalities. (English, French summary) Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291--362, Sémin. Congr., 1, Soc. Math. France, Paris, 1996.
  • Gromov, M. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999.
  • Katz, M. Systolic geometry and topology. With an appendix by J. Solomon. Mathematical Surveys and Monographs, volume 137. American Mathematical Society, 2007.
  • Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Ergo. Th. Dynam. Sys. 25 (2005), 1209-1220.
  • Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399-422. Available at arXiv:math.DG/0505007
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