Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length 2 by a 2-dimensional disk with the strongly isometric property, the round hemisphere has the least area. To explain this formulation, we start with the observation that the equatorial circle of the unit 2-sphere S^2 subset mathbb R^3 is a Riemannian circle S1 of length 2 . More precisely, the Riemannian distance function of S1 is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is only 2, whereas in the Riemannian circle it is .

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