Abstract：

The systole of a closed Riemannian manifold is defined to be the length of a shortest noncontractible loop. Gromov's systolic inequality relates systole to volume, which is a curvature free inequality. Gromov proved that systolic inequality holds on closed essential manifolds. Gromov's further work indicates that systolic inequality is related to topological complicatedness of manifolds. Analogously, Berger's embolic inequality is another curvature free inequality, also reflecting topological properties. In this talk, we introduce some new developments concerning the relation between these two curvature free inequalities and the topology of manifolds.