Systole (mathematics)
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Systole, when used as a term in mathematics, is used in Riemannian geometry. See Glossary of Riemannian and metric geometry.
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1. Systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner, and later developed by Mikhael Gromov (Mikhail Gromov) and others, in its arithmetic, ergodic, and topological manifestations.
2. 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. When X is a graph, the invariant is usually referred to as the GIRTH, ever since the 1947 article by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late 1940's, 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 the venerable 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. A systolic geometry and topology article list at Systolic bibliography currently contains over 120 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".
3. FLAVOR. To give a preliminary idea of the flavor of the field, one could make the following observations. The main thrust of Thom's remark to Berger quoted above, appears to be the following. Whenever one encounters an inequality relating geometric invariants, such a phenomenon in itself is interesting; all the more so when the inequality is sharp (i.e. optimal). The elegance of the inequalities of Loewner, Pu, and Gromov is indisputable. In systolic questions about surfaces, integral-geometric identities play a particularly important role. Roughly speaking, there is an integral identity relating area on the one hand, and an average of energies of a suitable family of loops, on the other. By the Cauchy-Schwarz inequality, energy is an upper bound for length squared, hence one obtains an inequality between area and the square of the systole. Such an approach works both for the Loewner inequality for the torus, and for Pu's inequality for the real projective plane. A number of new inequalities of this type have recently been discovered, including universal volume lower bounds.
4. The deepest result in the field is GROMOV'S INEQUALITY for the homotopy 1-systole of an essential n-manifold M. A summary of a proof, based on recent results in geometric measure theory by S. Wenger, building upon earlier work by L. Ambrosio and B. Kirchheim, appears in Chapter 12 of the book "Systolic geometry and topology" (Mathematical Surveys and Monographs, volume 137, American Mathematical Society, 2007). A completely different approach was recently proposed by L. Guth.
5. GROMOV'S STABLE SYSTOLIC INEQUALITY. A significant difference between 1-systolic invariants (defined in terms of lengths of loops) and the higher, k-systolic invariants (defined in terms of areas of cycles, etc.) should be kept in mind. While a number of optimal systolic inequalities, involving the 1-systoles, have by now been obtained, just about the only optimal inequality involving purely the higher k-systoles is M. Gromov's optimal stable 2-systolic inequality for complex projective space, where the optimal bound is attained by the Fubini-Study metric. Just how exceptional Gromov's stable inequality is, only became clear recently. Namely, we discovered that, contrary to expectation, the symmetric metric on the quaternionic projective plane is NOT its systolically optimal metric, in contrast with the complex case.
6. Similarly, just about the only nontrivial LOWER bound for a k-systole with k=2, results from recent work in gauge theory and J-holomorphic curves. The study of lower bounds for the conformal 2-systole of 4-manifolds has led to a simplified proof of the density of the image of the period map, by Jake Solomon.
7. SCHOTTKY PROBLEM. Perhaps one of the most striking applications of systoles is in the context of the Schottky problem, by P. Buser and P. Sarnak, who distinguished the Jacobians of Riemann surfaces among principally polarized abelian varieties, laying the foundation for systolic arithmetic .
8. Asking systolic questions often stimulates questions in related fields. Thus, a notion of "systolic category" of a manifold has been defined and investigated, exhibiting a connection to the LUSTERNIK-SCHNIRELMANN (LS) category. Note that the systolic category (as well as the LS category) is, by definition, an integer. Once the connection is established, the influence is mutual: known results about LS category stimulate systolic questions, and vice versa.
9. Asymptotic phenomena for the systole of surfaces of large genus have been shown to be related to interesting ERGODIC phenomena, and to properties of congruence subgroups of ARITHMETIC GROUPS. A bibliography for systoles in HYPERBOLIC GEOMETRY currently numbers dozens of articles. Gromov's filling area conjecture has been proved in a hyperelliptic setting. Other systolic ramifications of HYPERELLIPTICITY have been identifed in genus 2.
11. The SURVEYS in the field include M. Berger's survey (1993), as well as M. Gromov's survey (1996), as well as Gromov's book (1999), and survey by C. Croke and M. Katz (2003), as well as Katz's book (2007). These references may help a beginner enter the field. They also contain open problems to work on.
Berger, Marcel. 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.
Gromov, Mikhael. 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, Misha. 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.
Croke, Christopher B.; Katz, Mikhail. Universal volume bounds in Riemannian manifolds. Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 109--137, Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003.
Katz, Mikhail. Systolic geometry and topology. Mathematical Surveys and Monographs, volume 137. American Mathematical Society, 2007.
