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TABLEAUX OVER LIE ALGEBRAS, INTEGRABLE SYSTEMS, AND CLASSICAL SURFACE THEORY
, 2006
"... Abstract. Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfa ..."
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Abstract. Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G0system of Terng and the curved flat system of Ferus–Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss–Codazzi equations are associated with tableaux over sl(4, R) are discussed. 1.
CL.: Conformally flat submanifolds in spheres and integrable systems. arXiv:mathdg/0803.2754v2
, 2008
"... ABSTRACT. É. Cartan proved that conformally flat hypersurfaces in Sn+1 for n> 3 have at most two distinct principal curvatures and locally envelop a oneparameter family of (n − 1)spheres. We prove that the GaussCodazzi equation for conformally flat hypersurfaces in S4 is a soliton equation, and ..."
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Cited by 3 (3 self)
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ABSTRACT. É. Cartan proved that conformally flat hypersurfaces in Sn+1 for n> 3 have at most two distinct principal curvatures and locally envelop a oneparameter family of (n − 1)spheres. We prove that the GaussCodazzi equation for conformally flat hypersurfaces in S4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S4 and their loop group symmetries. We also generalise these results to conformally flat nimmersions in (2n − 2)spheres with flat and nondegenerate normal bundle. 1.
Goertsches, Generators for rational loop groups and geometric applications, arXiv:0803.0029v1 [math.DG
"... Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simpl ..."
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Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neoclassical groups and most of their symmetric spaces. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G2/SO(4) to which our theory applies. Contents
A CLASS OF OVERDETERMINED SYSTEMS DEFINED BY TABLEAUX: INVOLUTIVENESS AND CAUCHY PROBLEM
, 2006
"... Abstract. This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux. 1. ..."
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Abstract. This article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux. 1.
DIFFERENTIAL SYSTEMS ASSOCIATED WITH TABLEAUX OVER LIE ALGEBRAS
, 2007
"... We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [33]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiv ..."
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We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [33]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.
Research Statement
"... My research is in the area of differential geometry, centering around various applications of integrable systems to submanifold geometries. I will first explain the overall goal of this very interdisciplinary and active field, then summarize my contributions and some ongoing and future projects. 1 ..."
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My research is in the area of differential geometry, centering around various applications of integrable systems to submanifold geometries. I will first explain the overall goal of this very interdisciplinary and active field, then summarize my contributions and some ongoing and future projects. 1
ISOTHERMIC HYPERSURFACES IN R n+1
, 809
"... Abstract. A diagonal metric Pn i=1 giidx2i is termed Guichardk if Pn−k Pn i=n−k+1 gii = 0. A hypersurface in Rn+1 is isothermick if it admits line of curvature coordinates such that its induced metric is Guichardk. Isothermic1 surfaces in R 3 are the classical isothermic surfaces in R 3. Both isoth ..."
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Abstract. A diagonal metric Pn i=1 giidx2i is termed Guichardk if Pn−k Pn i=n−k+1 gii = 0. A hypersurface in Rn+1 is isothermick if it admits line of curvature coordinates such that its induced metric is Guichardk. Isothermic1 surfaces in R 3 are the classical isothermic surfaces in R 3. Both isothermick hypersurfaces in R n+1 and Guichardk orthogonal coordinate systems on R n are invariant under conformal transformations. A sequence of n isothermick hypersurfaces in R n+1 (Guichardk orthogonal coordinate systems on R n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal coordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichardk orthogonal coordinate systems on R n O(2n−k,k) and solutions of thesystem, and a correO(n)×O(n−k,k) spondence between Combescure sequences of isothermick hypersurfaces in R n+1 O(2n+1−k,k) and solutions of thesystem, both being inteO(n+1)×O(n−k,k) grable systems. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries. 1.