By Mikhail G. Katz

The systole of a compact metric house $X$ is a metric invariant of $X$, outlined because the least size of a noncontractible loop in $X$. whilst $X$ is a graph, the invariant is mostly often called the girth, ever because the 1947 article by way of 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 terms of the two-dimensional torus and genuine projective aircraft, respectively. presently, systolic geometry is a speedily constructing box, which reviews systolic invariants of their relation to different geometric invariants of a manifold. This e-book offers the systolic geometry of manifolds and polyhedra, beginning with the 2 classical inequalities, after which continuing to fresh effects, together with an evidence of M. Gromov's filling quarter conjecture in a hyperelliptic atmosphere. It then offers Gromov's inequalities and their generalisations, in addition to asymptotic phenomena for systoles of surfaces of huge genus, revealing a hyperlink either to ergodic thought 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 classification