Gromov's systolic inequality for essential manifolds
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In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;[1] it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.
Technically, let M be an essential Riemannian manifold of dimension n; denote by sysπ1(M) the homotopy 1-systole of M, that is, the least length of a non-contractible loop on M. Then Gromov's inequality takes the form
where Cn is a universal constant only depending on the dimension of M.