Systolic geometry

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.Definition: The cover M ¯ {displaystyle {ar {M}}} of the manifold M corresponding the subgroup Ker(φ) ⊂ π is called the universal (or maximal) free abelian cover.Definition: The Jacobi variety (Jacobi torus) of M is the torus J1(M)= H1(M,R)/H1(M,Z)RDefinition: The Abel–Jacobi map A M : M → J 1 ( M ) , {displaystyle A_{M}:M o J_{1}(M),} is obtained from the map above by passing to quotients. The Abel–Jacobi map is unique up to translations of the Jacobi torus. In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. 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 (i.e. a loop that cannot be contracted to a point in the ambient space X). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of X. When X is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student Pao Ming 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 (see reference below). A bibliography at the Website for systolic geometry and topology currently contains over 160 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently (see the 2006 paper by Katz and Rudyak below), the link with the Lusternik–Schnirelmann category has emerged. The existence of such a link can be thought of as a theorem in systolic topology. Every convex centrally symmetric polyhedron P in R3 admits a pair of opposite (antipodal) points and a path of length L joining them and lying on the boundary ∂P of P, satisfying An alternative formulation is as follows. Any centrally symmetric convex body of surface area A can be squeezed through a noose of length π A {displaystyle {sqrt {pi A}}} , with the tightest fit achieved by a sphere. This property is equivalent to a special case of Pu's inequality (see below), one of the earliest systolic inequalities. 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 classical isoperimetric inequality is a good example. 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, where the case of equality is attained by the flat torus whose deck transformations form the lattice of Eisenstein integers,