Alternatively, every metric on the sphere invariant under the antipodal map admits a pair of opposite points at Riemannian distance satisfying
A more detailed explanation of this viewpoint may be found at the page Introduction to systolic geometry.
Filling area conjecture
An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length by a -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 -sphere is a Riemannian circle of length . More precisely, the Riemannian distance function of 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 , whereas in the Riemannian circle it is .
We consider all fillings of by a -dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian metric of a circle of length . The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.
Pu's inequality bears a curious resemblance to the classical isoperimetric inequality
for Jordan curves in the plane, where is the length of the curve while is the area of the region it bounds. Namely, in both cases a 2-dimensional quantity (area) is bounded by (the square of) a 1-dimensional quantity (length). However, the inequality goes in the opposite direction. Thus, Pu's inequality can be thought of as an "opposite" isoperimetric inequality.
- Filling area conjecture
- Gromov's systolic inequality for essential manifolds
- Gromov's inequality for complex projective space
- Loewner's torus inequality
- Systolic geometry
- Systoles of surfaces
- Gromov, Mikhael (1983). "Filling Riemannian manifolds". J. Differential Geom. 18 (1): 1–147. MR 697984.
- Gromov, Mikhael (1996). "Systoles and intersystolic inequalities". In Besse, Arthur L. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) [Proceedings of the Roundtable on Differential Geometry]. Séminaires et Congrès. 1. Paris: Soc. Math. France. pp. 291–362. ISBN 2-85629-047-7. MR 1427752.
- Gromov, Misha (1999) . Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. 152. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Boston, MA: Birkhäuser Boston, Inc. ISBN 0-8176-3898-9. MR 1699320.
- Katz, Mikhail G. (2007). Systolic geometry and topology. Mathematical Surveys and Monographs. 137. With an appendix by J. Solomon. Providence, RI: American Mathematical Society. doi:10.1090/surv/137. ISBN 978-0-8218-4177-8. MR 2292367.
- Pu, Pao Ming (1952). "Some inequalities in certain nonorientable Riemannian manifolds". Pacific J. Math. 2 (1): 55–71. MR 0048886.