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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.
GRASSMANN GEOMETRIES IN INFINITE DIMENSIONAL HOMOGENEOUS SPACES AND AN APPLICATION TO REFLECTIVE SUBMANIFOLDS
, 2007
"... Abstract. Let U be a real form of a complex semisimple Lie group, and (τ, σ) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This giv ..."
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Abstract. Let U be a real form of a complex semisimple Lie group, and (τ, σ) a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. This gives many new examples of submanifolds as integrable systems. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space, U/L, one projects to. We apply the construction to a question which generalizes, to the context of reflective submanifolds of arbitrary symmetric spaces, the problem of isometric immersions of space forms with negative extrinsic curvature and flat normal bundle. For this problem, we prove that the only cases where local solutions exist are the previously known cases of space forms, in addition to our new example of constant curvature Lagrangian immersions into complex projective and complex hyperbolic spaces. We also prove nonexistence of global solutions in the compact case. For other reflective submanifolds, lower dimensional solutions exist, and can be described in terms of Grassmann geometries. We consider one example in detail, associated to the group G2, obtaining a special class of surfaces in S 6. 1.
LOOP GROUP DECOMPOSITIONS IN ALMOST SPLIT REAL FORMS AND APPLICATIONS TO SOLITON THEORY AND GEOMETRY
, 805
"... Abstract. We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action by the whole subgroup of loops which extend holomorphically to the exterior discon the ..."
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Cited by 4 (0 self)
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Abstract. We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action by the whole subgroup of loops which extend holomorphically to the exterior discon the Uhierarchy of the ZSAKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrasstype representation for Lorentzian harmonic maps (1 + 1 wave maps) from surfaces into compact symmetric spaces. An “Iwasawatype ” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given. 1.
A LOOP GROUP FORMULATION FOR CONSTANT CURVATURE SUBMANIFOLDS OF PSEUDOEUCLIDEAN SPACE
, 2006
"... Abstract. We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudoRiemannian manifold Mm c,r, of dimension m, constant Gauss curvature c ̸ = 0, and signature r, into the pseudoEuclidean space Rm+k s, of signature s ≥ r. In fact t ..."
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Abstract. We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudoRiemannian manifold Mm c,r, of dimension m, constant Gauss curvature c ̸ = 0, and signature r, into the pseudoEuclidean space Rm+k s, of signature s ≥ r. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudoRiemannian sphere or hyperbolic space S m+k s or H m+k s, which have previously been studied. A simple formula is given for obtaining these immersions from those loop group maps. 1.
GRASSMANN GEOMETRIES AND INTEGRABLE SYSTEMS
, 804
"... Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann subma ..."
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Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from the recent article [2]. 1.
RESULTS RELATED TO GENERALIZATIONS OF HILBERT’S NONIMMERSIBILITY THEOREM FOR THE HYPERBOLIC PLANE
, 710
"... Abstract. We discuss generalizations of the wellknown theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanif ..."
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Abstract. We discuss generalizations of the wellknown theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (noncompact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space H n into a Euclidean space E n+k, n≥2, and the problem of Lagrangian isometric immersions of H n into C n, n ≥ 2. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not. 1.
CONSTANT GAUSSIAN CURVATURE SURFACES IN THE 3SPHERE VIA LOOP GROUPS
"... Abstract. In this paper we study constant positive Gauss curvature K surfaces in the 3sphere S3 with 0 < K < 1 as well as constant negative curvature surfaces. We show that the socalled normal Gauss map for a surface in S3 with Gauss curvature K < 1 is Lorentz harmonic with respect to the ..."
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Abstract. In this paper we study constant positive Gauss curvature K surfaces in the 3sphere S3 with 0 < K < 1 as well as constant negative curvature surfaces. We show that the socalled normal Gauss map for a surface in S3 with Gauss curvature K < 1 is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if K is constant. We give a uniform loop group formulation for all such surfaces withK 6 = 0, and use the generalized d’Alembert method to construct examples. This representation gives a natural correspondence between such surfaces with K < 0 and those with 0 < K < 1.