Geometry of Energy and Bienergy Variations between Riemannian Manifolds


  • Received : 2014.11.15
  • Accepted : 2015.05.13
  • Published : 2015.09.23


In this note, we extend the definition of harmonic and biharmonic maps via the variation of energy and bienergy between two Riemannian manifolds. In particular we present some new properties for the generalized stress energy tensor and the divergence of the generalized stress bienergy.


Energy variation;L-harmonic maps;L-biharmonic maps;Stress L-bi-energy tensors


  1. M. Ara, Geometry of F-harmonic maps, Kodai Math. J., 22(1999), 243-263.
  2. P. Baird, J. C. Wood, Harmonic morphisms between Riemannain manifolds, Clarendon Press Oxford 2003.
  3. Y. J. Chiang, f-biharmonic Maps between Riemannian Manifolds, Department of Mathematics, University of Mary Washington Fredericksburg, VA 22401, USA 2012.
  4. Y.-J. Chiang and H. Sun,Biharmonic maps on V-manifolds, Int. J. Math. Math. Sci., 27(8)(2001), 477-484.
  5. N. Course, f-harmonic maps which map the boundary of the domain to one point in the target, New York Journal of Mathematics, 13(2007), 423-435.
  6. A.M. Cherif and M. Djaa, On generalized f-harmonic morphisms, Comment. Math. Univ. Carolin., 55(1)(2014), 1727.
  7. M. Djaa and A. M. Cherif, On Generalized f-biharmonic Maps and Stress f-bienergy Tensor, Journal of Geometry and Symmetry in Physics JGSP, 29(2013), 65-81.
  8. M. Djaa, A. M. Cherif, K. Zegga S. Ouakkas, On the Generalized of f-harmonic and f-bi-harmonic Maps, International Electronic Journal of Geometry, 1(2012), 1-11.
  9. J. Eells, p-Harmonic and Exponentially Harmonic Maps, lecture given at Leeds University, June 1993.
  10. J. Eells and J.H. Sampson, Harmonic mappings of riemannian manifolds, Amer. J. Math., 86(1964), 109-160.
  11. S. Feng and Y. Han, Liouville Type Theorems of f-Harmonic Maps with Potential, Results in Mathematics, 66(2014), 43-64.
  12. G. Y. Jiang, 2-Harmonic Maps Between Riemannian Manifolds, Annals of Math., China, 7A(4)(1986), 389-402.
  13. E. Loubeau and C. Oniciuc. On the biharmonic and harmonic indices of the hopf map, Transactions of the american mathematical society., 359(2007), 5239-5256.
  14. E. Loubeau, S. Montaldo, And C. Oniciuc, the stress-energy tensor for biharmonic maps, arXiv:math/0602021v1 [math.DG] 1 Feb 2006.
  15. S. Ouakkas, R. Nasri and M. Djaa. On the f-harmonic and f-biharmonic maps, J. P. Journal. of Geom. and Top., 10(1)(2010), 11-27.
  16. Y. L. Ou, On f-harmonic morphisms between Riemannian manifolds, arxiv:1103.5687 (2011), Chinese Ann Math, series B (2014).

Cited by

  1. On the bi-harmonic maps with potential 2017,
  2. General f-harmonic morphisms vol.22, pp.2, 2016,
  3. On the p-harmonic and p-biharmonic maps vol.109, pp.3, 2018,