On two dimensional isosystolic inequalities
Let M be a closed surface. For any Riemannian metric g on M, let Area(g) be its area, and let sys(g) by the length of its shortest noncontractible closed geodesic. The isosystolic constant of M is defined to be \(\Sigma_ M=\inf Area(g)/sys(g)^ 2\) where the infimum is taken over all Riemannian metrics on M. The exact value of \(\Sigma_ M\) has been calculated when M is a projective plane, a torus, and a Klein bottle. Otherwise, Gromov has found estimates from below for \(\Sigma_ M\) in terms of the first Betti number of M. The techniques of this paper produce both upper and lower bounds for \(\Sigma_ M\) in terms of an isosystolic constant for finite graphs whose Euler number is -\(\gamma\) where \(\gamma\) is the genus of M. They also produce a lower bound on \(\Sigma_ M\) in terms of \(\gamma\) which unfortunately is not as good as Gromov's lower bound.
scientific article; zbMATH DE number 4046580
scientific article