Geometric Interpretation of Second Elliptic Integrable System

In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space $G/G_0$ can be embedded into the twistor space of the corresponding symmetric space $G/H$. Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift $J$ taking values in $G/G_0 \hookrightarrow \Sigma(G/H)$. We begin the paper by an example: $G/H=\R^4$. We study also the structure of 4-symmetric bundles over Riemannian symmetric spaces

Supersymmetric Harmonic Maps into Symmetric Spaces

We study supersymmetric harmonic maps from the point of view of integrable system. 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). In the end of the paper we show 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

Sigma models with a Wess-Zumino term in twistor spaces

We characterize the Riemannian manifolds whose the twistor space satisfies the geometric properties necessary to the existence of some sigma model with a Wess-Zumino term on this twistor space. We prove that these manifolds are space forms. Then we study the Riemannian manifolds for which there exists a subbundle of the twistor space which satisfies these geometric properties and prove that in most cases these manifolds are locally homogeneous. In our study, we are led to prove some theorems about metric connections with parallel curvature: we prove for example that a metric connection with parallel curvature and with restricted holonomy group $SO(n)$ must be the Levi-Civita connection and therefore the Riemannian manifold is a space form. We also propose a general method to study metric connections with parallel curvature

Systèmes intégrables intervenant en géométrie différentielle et en physique mathématique

date de rédaction ( dernière version): novembre 2005Dêpot de ma thèse à la commission des thèses de Paris 7: 22 Novembre 2005.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).Notre thèse est divisée en 2 chapitres indépendants correspondant chacun à un article. Dans le premier chapitre, nous définissons une notion de surfaces isotropes dans les octonions, i.e. sur lesquelles certaines formes symplectiques canoniques s'annulent. En utilisant le produit vectoriel dans O, nous définissons une application rho de la grassmanienne des plans de O dans la sphère de dimension 6. Cela nous permet d'associer à chaque surface Sigma de O une fonction rho_Sigma de la surface sur la sphère. Alors, nous montrons que les surfaces isotropes de O telles que cette fonction est harmonique sont solutions d'un système complètement intégrable. En utilisant les groupes de lacets, nous construisons une représentation de type Weierstrass de ces surfaces. Par restriction au corps des quaternions, nous retrouvons comme cas particulier les surfaces lagrangiennes hamiltoniennes stationnaires de R^4. Par restriction à Im(H), nous retrouvons les surfaces CMC de R^3. Dans le second chapitre, nous étudions les applications supersymétriques harmoniques définies sur R^{2|2} et à valeurs dans un espace symétrique, du point de vue des systèmes intégrables. Il est bien connu que les applications harmoniques de R^2 à valeurs dans un espace symétrique sont solutions d'un système intégrable. Nous montrons que les applications superharmoniques de R^{2|2} dans un espace symétrique sont solutions d'un système intégrable, et que l'on a une représentation de type Weierstrass en termes de potentiels holomorphes (ainsi qu'en termes de potentiels méromorphes). Nous montrons également que les applications supersymétriques primitives de R^{2|2} dans un espace 4-symétrique donnent lieu, par restriction à R^2, à des solutions du système elliptique du second ordre associé à l'espace 4-symétrique considéré (au sens de C.L. Terng).Ceci nous permet d'obtenir, de manière conceptuelle, une sorte d'interprétation supersymétrique de tous les systèmes elliptiques du second ordre associés à un espace 4-symétrique, en particulier du système intégrable construit au chapitre 1 (et plus particulièrement des surfaces lagrangiennes hamiltoniennes stationnaires dans un espace symétrique)

Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

In this paper, we study all the elliptic integrable systems, in the sense of C.L. Terng [65]. That is to say the family of all the m-th elliptic integrable systems associated to a k ′-symmetric space N = G/G0. Here m ∈ N and k ′ ∈ N ∗ are integers. For example, it is known that the first elliptic integrable system associated to a symmetric space (resp. to a Lie group) is the equation for harmonic maps into this symmetric space (resp. this Lie group). Indeed it is well known that this harmonic maps equation can be written as a zero curvature equation: dαλ+ 1 2 [αλ∧αλ] = 0, ∀λ ∈ C ∗ , where αλ = λ −1 α ′ 1+α0+λα ′′ 1 is a 1-form on a Riemann surface L taking values in the Lie algebra g. This 1-form αλ is obtained as follows. Let f: L → N = G/G0 be a map from the Riemann surface L into the symmetric space G/G0. Then let F: L → G be a lift of f, and consider α = F −1.dF its Maurer-Cartan form. Then decompose α according to the symmetric decomposition g = g0 ⊕ g1 of g: α = α0 + α1. Finally, we define αλ: = λ −1 α ′ 1 + α0 + λα ′′ 1, ∀λ ∈ C ∗ , where α ′ 1,α ′′ 1 are the resp. the (1,0) and (0,1) parts o

Systèmes intégrables intervenant en géométrie différentielle et en physique mathématique

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