Annals of Global Analysis and Geometry

, Volume 32, Issue 3, pp 253–275 | Cite as

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

Original Paper

Abstract

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.

Keywords

Isometric immersions Space forms Pluriharmonic maps Loop groups 

Mathematics Subject Classification (2000)

Primary: 37K10 37K25 53C42 53B25 Secondary: 53C35 

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References

  1. 1.
    Barbosa J.L., Ferreira W., Tenenblat K. (1996) Submanifolds of constant sectional curvature in pseudo-Riemannian manifolds. Ann. Global Anal. Geom. 14: 381–401MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brander D., Dorfmeister J.: The generalized DPW method and an application to isometric immersions of space forms. arxiv preprint:0604247 (2006)Google Scholar
  3. 3.
    Burstall F.E., Ferus D., Pedit F., Pinkall U. (1993) Harmonic tori in symmetric spaces and commuting Hamlitonian systems on loop algebras. Ann. Math. 138 (2): 173–212MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen Q., Zuo D.F., Cheng Y. (2004) Isometric immersions of pseudo-Riemannian space forms. J. Geom. Phys. 52(3): 241–262MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dorfmeister J., Eschenburg J.H. (2003) Pluriharmonic maps, loop groups and twistor theory. Ann. Global Anal. Geom. 24: 301–321MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dorfmeister J., Pedit F., Wu H. (1998) Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6: 633–668MATHMathSciNetGoogle Scholar
  7. 7.
    Ferus D., Pedit F. (1996) Curved flats in symmetric spaces. Manuscripta Math. 91: 445–454MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ferus D., Pedit F. (1996) Isometric immersions of space forms and soliton theory. Math. Ann. 305: 329–342MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fordy, A.P., Wood, J.C. (eds.): Harmonic Maps and Integrable systems. Aspects of mathematics. Vieweg (1994)Google Scholar
  10. 10.
    Hilbert D. (1901) Ueber Flaechen von constanter Gaussscher Kruemmung. Trans. Amer. Math. Soc. 2: 87–99CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Li H. (1997) Global rigidity theorems of hypersurfaces. Ark. Mat. 35(2): 327–351MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moore J. (1972) Isometric immersions of space forms in space forms. Pacific J. Math. 40, 157–166MATHMathSciNetGoogle Scholar
  13. 13.
    Nikolayevsky Y. (1998) A non-immersion theorem for a class of hyperbolic manifolds. Differential Geom. Appl. 9, 239–242MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ohnita Y., Valli G. (1990) Pluriharmonic maps into compact lie groups and factorization into unitons. Proc. London Math. Soc. 61(3): 546–570MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pedit F. (1988) A nonimmersion theorem for spaceforms. Comment. Math. Helv. 63(4): 672–674MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pressley A., Segal G. (1986) Loop Groups. Oxford Math. monographs. Clarendon Press, OxfordGoogle Scholar
  17. 17.
    Tenenblat K., Terng C.L. (1980) Baecklund’s theorem for n-dimensional submanifolds of R 2n-1. Ann. Math. 111, 477–490CrossRefMathSciNetGoogle Scholar
  18. 18.
    Terng C.L. (1980) A higher dimensional generalisation of the sine-Gordon equation and its soliton theory. Ann. Math. 111, 491–510CrossRefMathSciNetGoogle Scholar
  19. 19.
    Toda, M.: Pseudospherical Surfaces via Moving Frames and Loop Groups. PhD Thesis. University of Kansas (2000)Google Scholar
  20. 20.
    Uhlenbeck K. (1989) Harmonic maps into lie groups: classical solutions of the chiral model. J. Differential Geom. 30, 1–50MATHMathSciNetGoogle Scholar
  21. 21.
    Wolf, J.: Spaces of Constant Curvature. Publish or Perish (1977)Google Scholar
  22. 22.
    Xavier F. (1985) A nonimmersion theorem for hyperbolic manifolds. Comment. Math. Helv., 60(2): 280–283MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKobe UniversityKobeJapan

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