Notion of Systole
The systole of a compact metric space is a metric invariant of, defined to be the least length of a noncontractible loop in . We will denote it as follows:
Note that a loop minimizing length is necessarily a closed geodesic. When 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 '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.
Read more about this topic: Introduction To Systolic Geometry
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