Systolic geometry

Summary

Systolic geometry studies the systole of a Riemannian manifold, the length of its shortest closed curve that cannot be shrunk to a point. Its main questions compare the systole with the volume of the manifold. Typical results say that a manifold with a long systole must have a large volume, as in Loewner's torus inequality and Gromov's systolic inequality. The field links Riemannian geometry with topology and with the geometry of numbers.