Sharif Digital Repository

In this thesis, we first define the pseudo-distance p on the group of Hamiltonian diffeomorphisms.Using the concept of displacement energy, we show that the pseudo-distance p is degenerate and if the manifold is closed, p will be zero for each p = 1; 2; 3; : : : Then, we introduce Lagrangian submanifolds and prove that if L R2n is a rational Lagrangian submanifold, we have the following inequality e(L) ≥ 1γ(L). : Finally, using the above inequality and the concept of displacement energy, for M = R2n we prove that 1 is non-degenerate. Therefore, the hypothesis for 1 to be a metric, are satisfied. This metric is called Hofer’s metric