Skip to main content
SpringerLink
Log in

Decompositions of para-complex vector bundles and para-complex affine immersions

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ∇ over a para-complex manifold. First we obtain results on the connections induced on the subbundles, their second fundamental forms and their curvature tensors. In particular we analyze para-holomorphic decompositions. Then we introduce the notion of para-complex affine immersions and apply the above results to obtain existence and uniqueness theorems for para-complex affine immersions. This is a generalization of the results obtained by Abe and Kurosu [AK] to para-complex geometry. Further we prove that any connection with vanishing (0, 2)-curvature, with respect to the grading defined by the para-complex structure, induces a unique para-holomorphic structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. Lond. A. 362 (1978), 425–461.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Abe and S. Kurosu, A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions, Result. Math. 44 (2003), 3–24.

    MathSciNet  MATH  Google Scholar 

  3. C.-L. Bejan, The existence problem of hyperbolic structures on vector bundles, Publ. Inst. Math. (Beograd), (N. S.) 53 [67] (1993), 133–138.

    MathSciNet  Google Scholar 

  4. V. Cortés, C. Mayer, T. Mohaupt and F. Saueressig, Special Geometry of Euclidean Supersymmetry I: Vectormultiplets, J. High Energy Phys. JHEP03 (2004) 028, hep-th/0312001 2004.

  5. F. Dillen, The affine differential geometry of complex hypersurfaces, Med. Konink. Acad. Wetensch. Belgie, 52 (1990), 89–112.

    MathSciNet  Google Scholar 

  6. F. Dillen and L. Vrancken, Complex affine hypersurfaces of ℂn+1, part 1, Bull. Soc. Math. Belg. Sér. B, 40 (1988), 245–271.

    Google Scholar 

  7. S. Erdem, Paraholomorphic structures and the connections of vector bundles over paracomplex manifolds, New Zealand J. of Math., Vol. 30 (2001), no.1, 41–50.

    MathSciNet  MATH  Google Scholar 

  8. K. Hasegawa, The fundamental theorems for affine immersions into hyperquadrics and its applications, Monatsh. Math., 131 (2000), 37–48.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami Shoten, Publishers and Princeton University Press, 1987.

  10. S. Kobayashi and K. Nomizu, Foundations of differential geometry, volume II, Interscience Publishers, 1969.

  11. K. Nomizu, U. Pinkall, F. Podesta, On the geometry of affine Kähler immersions, Nagoya Math. J., 120 (1990), 205–222.

    MathSciNet  MATH  Google Scholar 

  12. T. Okuda, On the fundamental theorems for purely real immersions, Master Thesis 1999, SUT.

  13. L. Schäfer, tt*-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Diff. Geom. and Appl. 24 no.1 (2006), 60–89.

    Article  MATH  Google Scholar 

  14. F. W. Warner, Foundations of differentiate manifolds and Lie groups, Scott, Foresman and Company (1971).

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Beringstraße 1, D-53115, Bonn, Germany

    Marie Amélie Lawn & Lars Schäfer

  2. Institut Élie Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239, F-54506, Vandœuvre-lès-Nancy Cedex, France

    Marie Amélie Lawn & Lars Schäfer

Authors
  1. Marie Amélie Lawn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars Schäfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marie Amélie Lawn.

Additional information

Research of the second author was supported by a grant of the ‘Studienstiftung des deutschen Volkes’.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lawn, M.A., Schäfer, L. Decompositions of para-complex vector bundles and para-complex affine immersions. Results. Math. 48, 246–274 (2005). https://doi.org/10.1007/BF03323368

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03323368

Keywords

MSC(2000)

Search

Navigation