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25
Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems
, 2000
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Geometries and symmetries of soliton equations and integrable elliptic systems
 IN SURVEYS ON GEOMETRY AND INTEGRABLE SYSTEMS, ADVANCED STUDIES IN PURE MATHEMATICS, MATHEMATICAL SOCIETY OF JAPAN NORTHEASTERN UNIVERSITY AND UC IRVINE EMAIL ADDRESS: TERNG@NEU.EDU MSRI, BERKELEY, CA 94720 EMAIL ADDRESS: EWANG@MRSI.ORG
, 2002
"... We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel are soliton flow ..."
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Cited by 26 (4 self)
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We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semisimple Lie algebra and finite order automorphisms. For example, the nonlinear Schrödinger equation, the nwave equation, and the sigmamodel are soliton flows; and the equation for harmonic maps from the plane to a compact Lie group, for primitive maps from the plane to a ksymmetric space, and constant mean curvature surfaces and isothermic surfaces in space forms are integrable elliptic systems. We also give a survey of • construction of solutions using loop group factorizations, • PDEs in differential geometry that are soliton equations or elliptic integrable systems, • similarities and differences of soliton equations and integrable elliptic
The Submanifold Geometries associated to Grassmannian Systems
 735, viii + 95
"... There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Gr ..."
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Cited by 12 (1 self)
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There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order nonlinear system of partial di#erential equations in n variables, the so called U/Ksystem. Let G m,n denote the Grassmannian of ndimensionallinear subspaces in R m+n , and G 1 m,n the Grassmannian of spacelike mdimensionallinear subspaces in the Lorentzian space R m+n,1 . In this paper, we use techniques from soliton theory to study submanifolds in space forms whose GaussCodazzi equations are gauge equivalent to the G m,n system or the G 1 m,n system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G m,n and G 1 m,n systems correspond toB acklund, Darboux and Ribaucour transformations for submanifolds. Tabl# of Contents 1.
The generalized DPW method and an application to isometric immersions of space forms
, 2008
"... Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S1 into G, of a suitable class. A differentiable map F from a manifold M into ΛG, is said to be of connection order ( b a) if the Fourier expansion in the loop parameter λ of the S1family of MaurerCartan forms for ..."
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Cited by 11 (9 self)
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Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S1 into G, of a suitable class. A differentiable map F from a manifold M into ΛG, is said to be of connection order ( b a) if the Fourier expansion in the loop parameter λ of the S1family of MaurerCartan forms for F, namely F −1 λ dFλ, is of the form Pb i=a αiλi. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order ( 1 −1) map, into a pair of simpler maps of order ( −1 −1) and (1 1) respectively. Conversely, one could construct such a harmonic map from any pair of ( −1 −1) and (1 1) maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order ( b a) map, for a < 0 < b, splits uniquely into a pair of (−1 a) and ( b 1) maps. As an application, we show that constant nonzero curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.
Isometric immersions of space forms and soliton theory
 Math. Ann
, 1996
"... The study of isometric immersions of space forms into space forms is a classical problem of differential geometry. In its simplest form it arises as the study of surfaces in 3space of constant (nonzero) Gaussian curvature. In this case the integrability condition reduces to the sin resp. sinhGord ..."
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Cited by 10 (1 self)
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The study of isometric immersions of space forms into space forms is a classical problem of differential geometry. In its simplest form it arises as the study of surfaces in 3space of constant (nonzero) Gaussian curvature. In this case the integrability condition reduces to the sin resp. sinhGordon equations. Due to the complicated structure of these equations one
Isothermic submanifolds of symmetric Rspaces
, 2009
"... We extend the classical theory of isothermic surfaces in conformal 3space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric Rspaces with essentially no loss of integrable structure. ..."
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Cited by 10 (5 self)
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We extend the classical theory of isothermic surfaces in conformal 3space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric Rspaces with essentially no loss of integrable structure.
Curved flats, pluriharmonic maps and constant curvature immersions into pseudoRiemannian space forms
, 2006
"... We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among ..."
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Cited by 9 (7 self)
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We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudoRiemannian space forms. As a corollary, we obtain a nonimmersibility theorem for spheres into certain pseudoRiemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.
Curved flats, exterior differential systems, and conservation laws, Complex, contact and symmetric manifolds
 235–254, Progr. Math., 234, Birkhauser
, 2005
"... Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. Th ..."
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Cited by 8 (3 self)
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Abstract. Let σ be an involution of a real semisimple Lie group U, U0 the subgroup fixed by σ, and U/U0 the corresponding symmetric space. Ferus and Pedit called a submanifold M of a rank r symmetric space U/U0 a curved flat if TpM is tangent to an rdimensional flat of U/U0 at p for each p ∈ M. They noted that the equation for curved flats is an integrable system. Bryant used the involution σ to construct an involutive exterior differential system Iσ such that integral submanifolds of Iσ are curved flats. Terng used r first flows in the U/U0hierarchy of commuting soliton equations to construct the U/U0system. She showed that the U/U0system and the curved flat system are gauge equivalent, used the inverse scattering theory to solve the Cauchy problem globally with smooth rapidly decaying initial data, used loop group factorization to construct infinitely many families of explicit solutions, and noted that many these systems occur as the GaussCodazzi equations for submanifolds in space forms. The main goals of this paper are: (i) give a review of these known results, (ii) use techniques from soliton theory to construct infinitely many integral submanifolds and conservation laws for the exterior differential system Iσ. 1.
Transformations of flat Lagrangian immersions and Egoroff nets, accepted
 Asian Journal of Mathematics
"... Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space U( ..."
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Cited by 7 (2 self)
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Abstract. We associate a natural λfamily (λ ∈ R \ {0}) of flat Lagrangian immersions in C n with nondegenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space U(n)⋉Cn O(n)⋉Rn. An interesting observation is that the family degenerates to an Egoroff net on R n when λ → 0. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The action of the generator with one simple pole gives the geometric Ribaucour transformation and we provide the permutability formula for such transformations. The action of the generator with two poles and the action of a rational loop in the translation subgroup produce new transformations. The corresponding results for flat Lagrangian submanifolds in CP n−1 and ∂invariant Egoroff nets follow nicely via a spherical restriction and Hopf fibration. 1.
The BianchiDarboux transform of Lisothermic surfaces
 Sch] [WP] W.K. Schief. Isothermic surfaces in spaces of
"... Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then ..."
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Cited by 7 (4 self)
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Abstract. We study an analogue of the classical Bäcklund transformation for Lisothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an Lisothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi–Darboux transforms. 1. Introduction. Certain types of integrable nonlinear PDEs (soliton equations) arise in differential geometry as compatibility conditions for the linear equations obeyed by frames adapted to surfaces in higher dimensional manifolds. In a number of situations, the construction of new solutions of the arising PDE relies on the existence of Bäcklund type