Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In Riemannian geometry, M. Gromov's systolic inequality for essential n-manifolds M dates from 1983. It is a lower bound for the volume of an arbitrary metric on M, in terms of its homotopy 1-systole. The homotopy 1- systole is the least length of a non-contractible loop. We will denote the homotopy 1-systole by the symbol operatorname{syspi}_1, so as to avoid confusion with the homology 1-systole.

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