Skip to Main content Skip to Navigation
Theses

Systèmes intégrables intervenant en géométrie différentielle et en physique mathématique

Abstract : Our thesis is divided into 2 independant chapters each other corresponding to a paper. In the first chapter, we define a notion of isotropic surfaces in O, i.e. on which some canonical symplectic forms vanish. Using the cross-product in O we define a map rho from the Grassmannian of plan of O to the 6 dimension sphere. This allows us to associate to each surface Sigma of O a function rho_Sigma. Then we show that the isotropic surfaces in O such that this function is harmonic are solutions of a completely integrable system. Using loop groups we construct a Weierstrass type representation of these surfaces.By restriction to the quaternions we obtain as a particular case the Hamiltonian Stationary Lagrangian surfaces of R^4, and by restriction to Im(H) we obtain the CMC surfaces of R^3. In the second chapter, we study supersymmetric harmonic maps from the point of view of integrable systems. It is well known that harmonic maps from R^2 into a symmetric space are solutions of a integrable system. We show here that the superharmonic maps from R^{2|2} into a symmetric space are solutions of a integrable system, more precisely of a first elliptic integrable system in the sense of C.L.Terng and that we have a Weierstrass-type representation in terms of holomorphic potentials (as well as of meromorphic potentials). We show also that superprimitive maps from R^{2|2} into a 4-symmetric space give us, by restriction to R^2, solutions of the second elliptic system associated to the previous 4-symmetric space. This allows us to obtain a kind of conceptual supersymmetric interpretation for any second elliptic system associated to a 4-symmetric space, in particular for the integrable system built in the first chapter (and more particular for Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces).
Document type :
Theses
Complete list of metadatas

Cited literature [23 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00277998
Contributor : Idrisse Khemar <>
Submitted on : Wednesday, May 7, 2008 - 4:50:13 PM
Last modification on : Thursday, December 10, 2020 - 10:50:33 AM
Long-term archiving on: : Friday, May 28, 2010 - 6:48:15 PM

Identifiers

  • HAL Id : tel-00277998, version 1

Citation

Idrisse Khemar. Systèmes intégrables intervenant en géométrie différentielle et en physique mathématique. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2006. Français. ⟨tel-00277998⟩

Share

Metrics

Record views

424

Files downloads

244