Pu's inequality
In 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. == Statement == A student of Charles Loewner, Pu proved in his 1950 thesis (Pu 1952) that every Riemannian surface M {\displaystyle M} homeomorphic to the real projective plane satisfies the inequality Area ( M ) ≥ 2 π Systole ( M ) 2 , {\displaystyle \operatorname {Area} (M)\geq {\frac {2}{\pi }}\operatorname {Systole} (M)^{2},} where Systole ( M ) {\displaystyle \operatorname {Systole} (M)} , the systole of M {\displaystyle M} , is the length of the shortest loop in M that cannot be contracted to a point in the ambient space X. The equality is attained precisely when the metric has constant Gaussian curvature.