Skip to main content
SpringerLink
Log in

Harmonic Almost Complex Structures with Respect to General Natural Metrics

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

Abstract

We continue here the study initiated in [9] on the harmonicity of certain geometric objects on the total space TM of the tangent bundle of a Riemannian space form (M(c), g). Precisely, in this paper we find all the general natural metrics on TM, with respect to which the canonical almost complex structure J on TM is harmonic. We also study the harmonicity of this tensor field with respect to the natural diagonal metrics. In particular, we obtain that J is harmonic with respect to the Sasaki metric on TM if and only if the base manifold is flat.

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. Abbassi MTK, Sarih M: On Riemannian g-natural metrics of the form a.g s + b.g h + c.g v on the tangent bundle of a Riemannian manifold (M, g). Mediterr. J. Math 1(2), 19–43 (2005)

    Article  MathSciNet  Google Scholar 

  2. Abbassi MTK, Sarih M: On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds. Diff. Geom. And its Appl 22, 19–47 (2005)

    MATH  MathSciNet  Google Scholar 

  3. Baird, P, Bejan, CL: Quasi-harmonic maps between almost symplectic manifolds, Harmonic morphisms, harmonic maps, and related topics (Brest, 1997), 75-95, Chapman and Hall/CRC Res. Notes Math., 413, Chapman and Hall/CRC, Boca Raton, FL. (2000)

  4. Baird, P, Wood, JC: Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monograph (New Series), Oxford University Press. (2003)

  5. Balmuş A, Oniciuc C, Papaghiuc N: Harmonic properties on the tangent bundle. An. Univ. Vest Timisoara, Ser. Mat.-Inform 42, 17–28 (2004)

    MATH  Google Scholar 

  6. Bejan CL, Benyounes M: Harmonic φ-morphisms. BeitrGe Algebra Geom 44(2), 309–321 (2003)

    MATH  MathSciNet  Google Scholar 

  7. Bejan CL, Benyounes M, Binh TQ: minimality, conformality, in terms of horizontal and vertical Lee forms. Publ. Math. Debrecen 59(3–4), 389–406 (2001)

    MATH  MathSciNet  Google Scholar 

  8. Bejan CL, Binh TQ, Tamssy L: Isometric or harmonic mappings of complete Riemannian manifolds. Publ. Math. Debrecen 60(3–4), 455–461 (2002)

    MATH  MathSciNet  Google Scholar 

  9. Bejan CL, Druţă-Romaniuc SL: Connections which are harmonic with respect to general natural metrics. Diff. Geom. Appl 30(4), 306–317 (2012)

    Article  MATH  Google Scholar 

  10. Calvaruso G, Perrone D: Harmonic morphisms and Riemannian geometry of tangent bundles. Ann. Global Anal. Geom. 39(2), 187–213 (2011)

    MATH  MathSciNet  Google Scholar 

  11. Druţă SL: Classes of General Natural Almost Anti-Hermitian Structures on the Cotangent Bundles. Mediterr J. Math 8, 161–179 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Druţă S, Oproiu V: General natural Kähler structures of constant holomorphic sectional curvature on tangent bundles. An. Şt. Univ. “Al.I.Cuza” Iaşi, Mat., N.S 53(1)1:49–166 (2007)

  13. Duggal KL, Ianuş S, Pastore AM: Maps interchanging f-structures and their harmonicity. Acta Appl. Math 67(1), 91–115 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Eells J, Sampson JH: Harmonic mappings of Riemannian manifolds. Amer. J. Math 86, 109–160 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garcia-Rio E, Vanhecke L, Vazquez-Abal ME: Harmonic endomorphism fields. Illinois J. Math 41(1), 23–30 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Grifone, J: Structure presquetangente et connexions I.;II.. Ann. Inst. Fourier (Grenoble) 22 , 1, 287–334, 22 (1972), 3, 291–338 (1972)

  17. Ianuş S, Pastore AM: Harmonic maps on contact metric manifolds. Ann. Math. Blaise Pascal 2(2), 43–53 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ianuş, S, Pastore, AM: Harmonic maps and f-structures with parallelizable kernel. New developments in differential geometry, Budapest 1996, 143–154, Kluwer Acad. Publ., Dordrecht, 1999.

  19. Ianuş, S, Pastore, AM: Harmonic maps and morphisms on metric f -manifolds with parallelizable kernel. Harmonic morphisms, harmonic maps, and related topics (Brest, 1997), 67-73, Chapman and Hall/CRC Res. Notes Math., 413, Chapman and Hall/CRC, Boca Raton, FL, 2000

  20. Ianuş S, Pastore AM: Some foliations and harmonic morphisms. Rev. Roumaine Math. Pures Appl. 50(5–6), 671–676 (2005)

    MATH  MathSciNet  Google Scholar 

  21. Ianuş S, Pastore AM: Some remarkable structures on Riemannian manifolds and harmonic maps. Tensor (N.S.) 69, 179–189 (2008)

    MATH  MathSciNet  Google Scholar 

  22. Ishihara S: Harmonic sections of tangent bundles. J. Math. Tokushima Univ 13, 23–27 (1979)

    MATH  MathSciNet  Google Scholar 

  23. Janys̆ka J: Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannan manifold. Archivum Mathematicum (Brno) 37, 143–160 (2001)

    MathSciNet  Google Scholar 

  24. Kolář, I, Michor, P, Slovak, J: Natural Operations in Differential Geometry, Springer Verlag, Berlin, 1993, vi, 434 pp

  25. Kowalski O, Sekizawa M: Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification. Bull. Tokyo Gakugei Univ 40(4), 1–29 (1988)

    MathSciNet  Google Scholar 

  26. Kowalski O, Sekizawa M: On curvatures of linear frame bundles with naturally lifted metrics. rend. Sem. Mat. Univ. Pol. Torino 63(3), 283–296 (2005)

    MATH  MathSciNet  Google Scholar 

  27. Mok KP, Patterson EM, Wong YC: Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle. Trans. Amer. Math. Soc 234, 253–278 (1977)

    MATH  MathSciNet  Google Scholar 

  28. Munteanu MI: Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold. Mediterranean Journal of Mathematics 5(1), 43–59 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nouhaud, O: Applications harmoniques d’une variété riemanniene dans son fibré tangent. (1977)

  30. Oniciuc, C: Tangency and Harmonicity Properties. Ph.D. Thesis, DGDS Monographs 3, Geometry Balkan Press, Bucureşti, 2003.

  31. Oproiu V: Harmonic maps between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino 47, 47–55 (1989)

    MATH  MathSciNet  Google Scholar 

  32. Oproiu V: On the harmonic sections of cotangent bundles. Rend. Sem. Fac. Sci. Univ. Cagliari 59(2), 177–184 (1989)

    MATH  MathSciNet  Google Scholar 

  33. Oproiu V: A generalization of natural almost Hermitian structures on the tangent bundles. Math. J. Toyama Univ 22, 1–14 (1999)

    MATH  MathSciNet  Google Scholar 

  34. Oproiu V: Some new geometric structures on the tangent bundles. Publ. Math. Debrecen 55(3-4), 261–281 (1999)

    MATH  MathSciNet  Google Scholar 

  35. Oproiu V, Papaghiuc N: General natural Einstein Kahler structures on tangent bundles. Diff. Geom. Appl 27, 384–392 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Piu MP: Campi di vettori ed applicazioni armoniche. Rend. Sem. Fac. Sci. Univ. Cagliari 52, 85–94 (1982)

    MATH  MathSciNet  Google Scholar 

  37. Sasaki S: On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J 10, 338–354 (1958)

    Article  MATH  Google Scholar 

  38. Tahara M, Watanabe Y: Natural almost Hermitian, Hermitian and Kähler metrics on the tangent bundles. Math. J. Toyama Univ 20, 149–160 (1997)

    MATH  MathSciNet  Google Scholar 

  39. Terng CL: Natural vector bundles and natural differential operators. Am. J. Math 100, 775–828 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  40. Tondeur Ph, Vanhecke L: Harmonicity of a foliation and of an associated map. Bull. Austral. Math. Soc 54(22), 41–246 (1996)

    MathSciNet  Google Scholar 

  41. Yano, K, Ishihara, S: Tangent and Cotangent Bundles M. Dekker Inc., New York 1973

Download references

Author information

Authors and Affiliations

  1. Universitatea Tehnica “Gheorghe Asachi” Iaşi, Iaşi, Romania

    Cornelia-Livia Bejan

  2. Seminarul Matematic, Universitatea “Alexandru Ioan Cuza” din Iaşi Bd. Carol I, No. 1, 700506, Iaşi, Romania

    Cornelia-Livia Bejan

  3. Faculty of Mathematics, Universitatea “Alexandru Ioan Cuza” din Iaşi Bd. Carol I, No. 1, 700506, Iaşi, Romania

    Simona-Luiza Druţă-Romaniuc

Authors
  1. Cornelia-Livia Bejan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simona-Luiza Druţă-Romaniuc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cornelia-Livia Bejan.

Additional information

Dedicated to the memory of Professor Mircea Craioveanu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bejan, CL., Druţă-Romaniuc, SL. Harmonic Almost Complex Structures with Respect to General Natural Metrics. Mediterr. J. Math. 11, 123–136 (2014). https://doi.org/10.1007/s00009-013-0302-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-013-0302-0

Mathematics Subject Classification (2000)

Keywords

Search

Navigation