The systole of a compact metric house $X$ is a metric invariant of $X$, outlined because the least size of a noncontractible loop in $X$. while $X$ is a graph, the invariant is mostly known as the girth, ever because the 1947 article via W. Tutte. the 1st nontrivial effects for systoles of surfaces are the 2 classical inequalities of C. Loewner and P. Pu, counting on integral-geometric identities, in relation to the two-dimensional torus and genuine projective airplane, respectively. at the moment, systolic geometry is a speedily constructing box, which reports systolic invariants of their relation to different geometric invariants of a manifold. This booklet provides the systolic geometry of manifolds and polyhedra, beginning with the 2 classical inequalities, after which continuing to fresh effects, together with an explanation of M. Gromov's filling zone conjecture in a hyperelliptic environment. It then provides Gromov's inequalities and their generalisations, in addition to asymptotic phenomena for systoles of surfaces of huge genus, revealing a hyperlink either to ergodic concept and to houses of congruence subgroups of mathematics teams. the writer comprises effects at the systolic manifestations of Massey items, in addition to of the classical Lusternik-Schnirelmann type.
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