Systoles and intersystolic inequalities. (English) Zbl 0877.53002

Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 291-362 (1996).
This is a great overview of an active area in Riemannian geometry. The results surveyed herein are generalizations of an unpublished inequality discovered by Charles Loewner, circa 1949, which relates the length of the shortest noncontractible closed curve of an arbitrary Riemannian metric on the 2-torus to its surface area. The generalizations involve the \(k\)-dimensional systole of a Riemannian manifold, which can be described as the infimum of the \(k\)-dimensional volumes of the representatives of nonzero \(k\)-dimensional homology classes. Here, among many others, are inequalities relating the \(1\)-systole, area, and genus of a compact surface, inequalities involving \(1\)-systoles, \(2\)-systoles, and volumes in acyclic spaces, and even the calculation of \(k\)-systoles of projective spaces and their deformations. This is exciting mathematics, and this article, which includes interesting examples, open problems, and references to 60 papers, is a good place to find out about it.
For the entire collection see [Zbl 0859.00016].
Reviewer: J.Hebda (St.Louis)


53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C22 Geodesics in global differential geometry