The Geometry of the Group of Symplectic Diffeomorphisms

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Kalami Yazdi, Ali | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43740 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Eftekhary, Eiman; Esfahani Zadeh, Mostafa
Abstract:
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
Keywords:
Symplectic Manifold ; Symplectomorphism ; Hamiltonian Vector Field ; Hamiltonian Diffeomorphisms Group ; Hofer Metric ; Lagrangian Submanifold