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.
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. A closed manifold is called essential if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corresponding Eilenberg–MacLane space.
The starting point of the proof is the imbedding of X into the Banach space of Borel functions on X, equipped with the sup norm.
Guth (2011) and Ambrosio & Katz (2011) developed approaches to the proof of Gromov's systolic inequality for essential manifolds.
A uniform inequality for arbitrary 2-complexes with non-free fundamental groups is available, whose proof relies on the Grushko decomposition theorem.