Gromov's systolic inequality for essential manifolds

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; 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

${\displaystyle \left(\operatorname {sys\pi } _{1}(M)\right)^{n}\leq C_{n}\operatorname {vol} (M),}$

where C_n is a universal constant only depending on the dimension of M.