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- W2819405267 abstract "Let $G$ be a complex Lie group and $Lambda G$ denote the group of maps from the unit circle ${mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $Lambda G$, is said to be of emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $lambda$ of the ${mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely $F_lambda^{-1} dd F_lambda$, is of the form $sum_{i=a}^b alpha_i lambda^i$. 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 $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an application, we show that constant non-zero 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." @default.
- W2819405267 created "2018-07-19" @default.
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- W2819405267 date "2006-04-11" @default.
- W2819405267 modified "2023-09-27" @default.
- W2819405267 title "Generalized DPW method and an application to space forms" @default.
- W2819405267 hasPublicationYear "2006" @default.
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