Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms
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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 pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian 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.
KeywordsIsometric immersions Space forms Pluriharmonic maps Loop groups
Mathematics Subject Classification (2000)Primary: 37K10 37K25 53C42 53B25 Secondary: 53C35
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- 2.Brander D., Dorfmeister J.: The generalized DPW method and an application to isometric immersions of space forms. arxiv preprint:0604247 (2006)Google Scholar
- 9.Fordy, A.P., Wood, J.C. (eds.): Harmonic Maps and Integrable systems. Aspects of mathematics. Vieweg (1994)Google Scholar
- 16.Pressley A., Segal G. (1986) Loop Groups. Oxford Math. monographs. Clarendon Press, OxfordGoogle Scholar
- 19.Toda, M.: Pseudospherical Surfaces via Moving Frames and Loop Groups. PhD Thesis. University of Kansas (2000)Google Scholar
- 21.Wolf, J.: Spaces of Constant Curvature. Publish or Perish (1977)Google Scholar