Skip to main content
SpringerLink
Log in

Harmonicity and minimality of oriented distributions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS 4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to more general situations coming from 3-Sasakian and quaternionic geometry.

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. J. Berndt,Real hypersurfaces in quaternionic space forms, Journal für die reine und angewandte Mathematik419 (1991), 9–26.

    MATH  MathSciNet  Google Scholar 

  2. J. Berndt and Y. J. Suh,Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Mathematik127 (1999), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Berndt, L. Vanhecke and L. Verhóczki,Harmonic and minimal unit vector fields on Riemannian symmetric spacces, Illinois Journal of Mathematics47 (2003), 1273–1286.

    MATH  MathSciNet  Google Scholar 

  4. A. L. Besse,Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebie 3 Folge, Band 10, Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  5. D. E. Blair,Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics 203, Birkhäuser, Boston, 2002.

    MATH  Google Scholar 

  6. E. Boeckx and L. Vanhecke,Harmonic and minimal vector fields on tangent and unit it tangent bundles, Differential Geometry and its Applications13 (2000), 77–93.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Boeckx and L. Vanhecke,Harmonic and minimal radial vector fields, Acta Mathematica Hungarica90 (2001), 317–331.

    Article  MATH  MathSciNet  Google Scholar 

  8. E. Boeckx and L. Vanhecke,Isoparametric functions and harmonic and minimal unit vector fields, inGlobal Differential Geometry: The Mathematical Legacy of Alfred Gray (M. Fernández and J. A. Wolf, eds.), Contemporary Mathematics288, American Mathematical Society, Providence, RI, 2001, pp. 20–31.

    Google Scholar 

  9. P. M. Chacón, A. M. Naveira and J. M. Weston,On the energy of distributions, with application to the quaternionic Hopf fibration, Monatshefte für Mathematik133 (2001), 281–294.

    Article  MATH  Google Scholar 

  10. B-Y. Choi and J-W. Yim,Distributions on Riemannian manifolds, which are harmonic maps, Tôhoku Mathematical Journal55 (2003), 175–188.

    Article  MATH  MathSciNet  Google Scholar 

  11. O. Gil-Medrano,Relationship between volume and energy of vector fields, Differential Geometry and its Applications15 (2001), 137–152.

    Article  MATH  MathSciNet  Google Scholar 

  12. O. Gil-Medrano, J. C. González-Dávila and L. Vanhecke,Harmonic and minimal invariant unit vector fields on homogeneous Riemmannian manifolds, Houston Journal of Mathematics27 (2001), 377–409.

    MATH  MathSciNet  Google Scholar 

  13. O. Gil-Medrano and E. Llinares-Fuster,Minimal unit vector fields, Tôhoku Mathematical Journal54 (2002), 71–84.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. C. González-Dávila and L. Vanhecke,Examples of minimal unit vector fields, Annals of Global Analysis and Geometry18 (2000), 385–404.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. C. González-Dávila and L. Vanhecke,Minimal and harmonic charracteristics vector fields on three-dimensional contact metric manifolds, Journal of Geometry72 (2001), 65–76.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. C. González-Dávila and L. Vanhecke,Energy and volume of unit vector fields on three-dimensional Riemannian manifolds, Differential Geometry and its Applications16 (2002), 225–244.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. C. González-Dávila and L. Vanhecke,Invariant harmonic unit vector fields on Lie groups, Bollettino dell'Unione Matematica Italiana. Sezione B5 (2002), 377–403.

    MATH  Google Scholar 

  18. T. Kashiwada,On a contact 3-structure, Mathematische Zeitschrift238 (2002), 829–832.

    Article  MathSciNet  Google Scholar 

  19. J. L. KonderakOn sections of fibre bundles, which are harmonic maps, Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie (N.S.)90 (1999), 341–352.

    MathSciNet  Google Scholar 

  20. O. Kowalski,Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, Journal für die reine und angewandte Mathematik250 (1971), 124–129.

    Article  MATH  Google Scholar 

  21. L. Ornea and L. Vanhecke,Harmonicity and minimality of vector fields and distributions on locally conformal Kähler and hyperkähler manifolds, Bulletin of the Belgium Mathematical Society. Simon Stevin, to appear.

  22. M. Salvai,On the emergy of sections of trivializable sphere bundles, Rendiconti del Seminario Mathematico della Universitá e Politecnicoi di Torino60 (2002), 147–155.

    MATH  MathSciNet  Google Scholar 

  23. K. Tsukada and L. Vanhecke, Minimal and harmonic unit vector fields inG 2(ℂm+2) and its dual space, Monatscheft für Mathematik130 (2000), 143–154.

    Article  MATH  MathSciNet  Google Scholar 

  24. K. Tsukada and L. Vanhecke,Minimality and harmonicity for Hopf vector fields, Illinois Journal of Mathematics45 (2001), 441–451.

    MATH  MathSciNet  Google Scholar 

  25. G. Wiegmink,Total bending of vector fields on Riemannian manifolds, Mathematische Annalen303 (1995), 325–344.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. M. Wood,A class of harmonic almost-product structures, Journal of Geometry and Physics14 (1994), 25–42.

    Article  MATH  MathSciNet  Google Scholar 

  27. C. M. Wood,The energy of Hopf vector fields, Manuscripta Mathematica101 (2000), 71–88.

    Article  MATH  MathSciNet  Google Scholar 

  28. C. M. Wood,Harmonic sections ond homogeneous fibre bundles, Differential Geometry and its Applications19 (2003), 193–210.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Valencia, 46100 Burjassot, Valencia, Spain

    O. Gil-Medrano

  2. Departamento de Matemática Fundamental, Sección de Geometría y Topología, Universidad de La Laguna, La Laguna, Spain

    J. C. González-Dávila

  3. Department of Mathematics, Katholieke Universiteit Leuwen, Celestijnenlaan 200 B, 3001, Leuven, Belgium

    L. Vanhecke

Authors
  1. O. Gil-Medrano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. C. González-Dávila
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Vanhecke
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by DGI Grant No. BFM2001-3548.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gil-Medrano, O., González-Dávila, J.C. & Vanhecke, L. Harmonicity and minimality of oriented distributions. Isr. J. Math. 143, 253–279 (2004). https://doi.org/10.1007/BF02803502

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02803502

Keywords

Search

Navigation