Pu's inequality

In differential geometry, Pu's inequality is an inequality proved by Pao Ming Pu for the systole of an arbitrary Riemannian metric on the real projective plane RP².

A student of Charles Loewner's, P.M. Pu proved in a 1950 thesis (Pu 1952) that every metric on the real projective plane ${\displaystyle \mathbb {RP} ^{2}}$ satisfies the optimal inequality

where sys is the systole. The boundary case of equality is attained precisely when the metric is of constant Gaussian curvature.

Alternatively, every metric on the sphere ${\displaystyle S^{2}}$ invariant under the antipodal map admits a pair of opposite points ${\displaystyle p,q\in S^{2}}$ at Riemannian distance ${\displaystyle d=d(p,q)}$ satisfying ${\displaystyle d^{2}\leq {\frac {\pi }{4}}\operatorname {area} (S^{2}).}$

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