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Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 503–520 | Cite as

Moduli of Einstein–Hermitian harmonic mappings of the projective line into quadrics

  • Oscar Macia
  • Yasuyuki Nagatomo
Article

Abstract

The present article studies the class of Einstein–Hermitian harmonic maps of constant Kähler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image and gauge equivalence using the language of vector bundles and representation theory. It is shown that the dimension of the moduli spaces is independent of the Einstein–Hermitian constant and rigidity of the associated real standard, and totally real maps are examined. Finally, certain classical results concerning embeddings of two-dimensional spheres into spheres are rephrased and derived in our formalism.

Keywords

Moduli space Einstein–Hermitian connection Harmonic mapping Complex projective line Complex hyperquadric Grassmannian manifold 

Mathematics Subject Classification

53C07 58E20 

Notes

Acknowledgements

The first-named author would like to thank the hospitality of Meiji University where part of this work was developed. The work of the first-named author was supported by the Spanish Agency of Scienctific and Technological Research (DGICT) and FEDER project MTM20136961. The work of the second-named author was supported by JSPS KAKENHI Grant Number 17K05230.

References

  1. 1.
    Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^{2}\) into \({{\mathbf{C}}} {\rm P}^{n}\), Mathematische Annalen. 279(4): 599–620 (1987/88)Google Scholar
  2. 2.
    Calabi, E.: Isometric imbedding of complex manifolds. Ann. Math. 58, 1–23 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, B.Y., Ogiue, K.: On totally real submanifolds. Trans. Am. Math. Soc. 193, 257–266 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chern, S.S., Wolfson, J.: Minimal surfaces by moving frames. Am. J. Math. 105, 59–83 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    do Carmo, M.P., Wallach, N.R.: Minimal immersions of spheres into spheres. Ann. Math 93, 43–62 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eells, J., Lemaire, L.: A report on Harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Iwanami Shoten and Princeton University, Tokyo (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    Koga, I., Nagatomo, Y.: A study of submanifolds of the complex grassmannian manifold with parallel second fundamental form. Tokyo J. Math. 39(1), 173–185 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Macia, O., Nagatomo, Y., Takahashi, M.: Holomorphic isometric embeddings of the projective line into quadrics, Tohoku Math. J. (2) 69(4), 525–545 (2017). https://www.math.tohoku.ac.jp/tmj/Nissue.html
  10. 10.
    Nagatomo, Y.: Harmonic maps into Grassmannian manifolds, arXiv: mathDG/1408. 1504
  11. 11.
    Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Takeuchi, M.: Modern Spherical Functions, Translations of Mathematical Monographs, vol. 135. American Mathematical Society, Providence (1994)CrossRefGoogle Scholar
  13. 13.
    Wang, J., Jiao, X.: Totally real minimal surfaces in the complex hyperquadrics. Diff. Geom. and its Appl. 31, 540–555 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of ValenciaValenciaSpain
  2. 2.Department of MathematicsMeiji UniversityKawasaki-shiJapan

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