Pu's Inequality
Pu's inequality for the real projective plane applies to general Riemannian metrics on .
A student of Charles Loewner's, Pao Ming Pu proved in a 1950 thesis (published in 1952) that every metric on the real projective plane satisfies the optimal inequality
where is the systole. The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature. Alternatively, the inequality can be presented as follows:
There is a vast generalisation of Pu's inequality, due to Mikhail Gromov, called Gromov's systolic inequality for essential manifolds. To state his result, one requires a topological notion of an essential manifold.
Read more about this topic: Introduction To Systolic Geometry
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