Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

International Mathematics Research Notices

2007

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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 non-existence 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 .

Section: The Non-compact Casementioning

confidence: 94%

Section: The Non-compactmentioning

confidence: 99%

Grassmann Geometries in Infinite Dimensional Homogeneous Spaces and an Application to Reflective Submanifolds

International Mathematics Research Notices

2007

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“…The S 1 reality condition is of the second kind, and so leads to type ( b −b ) τ maps; a different example of a ( b −b ) τ map will be discussed below, in Sect. 6.…”

Section: Limited Connection Order Maps Into Loop Groupsmentioning

confidence: 99%

Generalized DPW method and an application to isometric immersions of space forms

Dorfmeister²

2008

Math. Z.

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Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S 1 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 S 1 -family of Maurer-Cartan forms for F, namely F −1 λ dF λ , is of the form b i=a α i λ i . 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 non-zero sectional 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. Mathematics Subject Classification (2000)Primary 37K10 · 37K25 · 53C42 · 53B25; Secondary 53C35 J. Dorfmeister was supported by DFG grant DO 776.

“…C is a subgroup of ΛGL(n, C), and, by assumption, ρ is extended to ΛG C by the formula (3). Hence this extension is also the restriction to ΛG C of the involutionρ discusses in the U (n) case.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…, where −1 < c, or into a sphere S n+k . Solutions to both of these problems are obtained from just one map into a loop group, as follows (for details not proved here, see [3]): let G C = SO(n + k + 1, C), where k ≥ n − 1, and define the following three involutions,σ,ρ andτ on ΛG C by the formulae:…”

Section: An Application Of the Theoremsmentioning

confidence: 99%

Loop group decompositions in almost split real forms and applications to soliton theory and geometry

Journal of Geometry and Physics

2008

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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 U -hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps (1 + 1 wave maps) from surfaces into compact symmetric spaces. An "Iwasawa-type" 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.