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Almost CR quaternionic manifolds and their immersibility in \(\mathbb {H}\)P\(^n\).

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Abstract

We apply the general theory of codimension one integrability conditions for G-structures developed in Santi (Ann Mat Pura Appl 195:1463–1489, 2016) to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.

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References

  1. Alekseevsky, D.A., Kamishima, Y.: Pseudo-conformal quaternionic CR structure on (4n+3)-dimensional manifolds. Ann. Mat. Pura Appl. 187, 487–529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. de Andrés, L.C., Fernández, M., Ivanov, S., Santisteban, J.A., Ugarte, L., Vassilev, D.: Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures. Ann. Mat. Pura Appl. 193, 261–290 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alekseevsky, D.A., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. 171, 205–273 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreotti, A., Hill, C.D.: Complex characteristic coordinates and tangential Cauchy-Riemann equations. Ann. Sc. Norm. Sup. Pisa 26, 299–324 (1979)

    MathSciNet  MATH  Google Scholar 

  5. Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque, vol. 265. Société Mathématique de France, Paris (2000)

  6. Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate Texts in Mathematics, vol. 82, Springer, New York (1982)

  7. Conti, D., Fernández, M., Santisteban, J.A.: On seven dimensional quaternionic contact solvable Lie groups. Forum Math. 26, 547–576 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duchemin, D.: Quaternionic-contact hypersurfaces. (2006). arXiv:math/0604147 [math.AG]

  9. Guillemin, V.: The integrability problem for \(G\)-structures. Trans. Am. Math. Soc. 116, 544–560 (1965)

    MathSciNet  MATH  Google Scholar 

  10. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Third Printing, Revised. Springer, New York, Berlin (1978)

    MATH  Google Scholar 

  11. Kobayashi, S.: Transformation Groups in Differential Geometry. Springer, New York, Heidelberg (1972)

    Book  MATH  Google Scholar 

  12. Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74, 329–387 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kuranishi, M.: Strongly pseudo-convex CR structures over small balls, Part III. Ann. Math. 116, 249–330 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Marchiafava, S., Ornea, L., Pantilie, R.: Twistor theory for CR quaternionic manifolds and related structures. Monatsh. Math. 167, 531–545 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ochiai, T.: Geometry associated with semisimple flat homogeneous spaces. Trans. Am. Math. Soc. 152, 159–193 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  16. Salamon, S.M.: Differential geometry of quaternionic manifolds. Ann. Sci. Ecole Norm. Sup. 19, 31–55 (1986)

    MathSciNet  MATH  Google Scholar 

  17. Santi, A.: A generalized integrability problem for \(G\)-structures. Ann. Mat. Pura Appl. 195, 1463–1489 (2016)

  18. Singer, I.M., Sternberg, S.: The infinite groups of Lie and Cartan I. The transitive groups. J. Analyse Math. 15, 1–114 (1965)

  19. Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, Inc., Englewood Cliffs (1964)

    MATH  Google Scholar 

  20. Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22, 413–494 (1993)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

Part of this work was done while the author was a post-doc at the University of Parma. The author would like to thank the Mathematics Department and in particular A. Tomassini and C. Medori for support and ideal working conditions.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Universitá di Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy

    Andrea Santi

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  1. Andrea Santi
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Correspondence to Andrea Santi.

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Communicated by Vicente Cortés.

This research was partially supported by the Project Firb 2012 Geometria differenziale e teoria geometrica delle funzioni, by GNSAGA of INdAM and by Project F1R-MTH-PUL-08HALO-HALOS08 of University of Luxembourg.

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Santi, A. Almost CR quaternionic manifolds and their immersibility in \(\mathbb {H}\)P\(^n\).. Abh. Math. Semin. Univ. Hambg. 87, 83–103 (2017). https://doi.org/10.1007/s12188-016-0136-3

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  • DOI: https://doi.org/10.1007/s12188-016-0136-3

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