Pu's inequalityIn differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it. A student of Charles Loewner, Pu proved in his 1950 thesis that every Riemannian surface homeomorphic to the real projective plane satisfies the inequality where is the systole of . The equality is attained precisely when the metric has constant Gaussian curvature.
Gromov's systolic inequality for essential manifoldsIn 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.
Loewner's torus inequalityIn differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. In 1949 Charles Loewner proved that every metric on the 2-torus satisfies the optimal inequality where "sys" is its systole, i.e. least length of a noncontractible loop. The constant appearing on the right hand side is the Hermite constant in dimension 2, so that Loewner's torus inequality can be rewritten as The inequality was first mentioned in the literature in .
Systolic geometryIn mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. The systole of a compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X (i.e.
