Annals of Global Analysis and Geometry

, Volume 39, Issue 2, pp 187–213 | Cite as

Harmonic morphisms and Riemannian geometry of tangent bundles

  • Giovanni CalvarusoEmail author
  • Domenico Perrone


Let (TM, G) and \({(T_1 M,\tilde G)}\) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and \({\pi_1:(T_1 M,\tilde G) \rightarrow (M,g)}\), we give necessary and sufficient conditions for π and π 1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection \({\Phi:(TM-\{0\},G)\to (T_1 M,\tilde G)}\) with respect to Riemannian g-natural metrics \({G,\tilde G}\) of Kaluza–Klein type.


Harmonic maps Harmonic morphisms Tangent and unit tangent sphere bundles Riemannian g-natural metrics 

Mathematics Subject Classification (2000)

58E20 53C43 


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  1. 1.
    Abbassi M.T.K., Calvaruso G.: g-Natural contact metrics on unit tangent sphere bundles. Monatsh. Math. 151, 89–109 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Abbassi M.T.K., Calvaruso G.: Harmonic maps having tangent bundles with g-natural metrics as source or target. Rend. Sem. Mat. Torino 68, 1–20 (2010)Google Scholar
  3. 3.
    Abbassi M.T.K., Kowalski O.: On Einstein Riemannian g-natural metrics on unit tangent sphere bundles. Ann. Global Anal. Geom. 38(1), 11–20 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Abbassi M.T.K., Kowalski O.: Naturality of homogeneous metrics on Stiefel manifolds SO(m + 1)/SO(m − 1). Diff. Geom. Appl. 28(2), 131–139 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Abbassi M.T.K., Sarih M.: On natural metrics on tangent bundles of Riemannian manifolds. Arch. Math.(Brno) 41, 71–92 (2005)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Abbassi M.T.K., Sarih M.: On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds. Diff. Geom. Appl. 22(1), 19–47 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Abbassi M.T.K., Calvaruso G., Perrone D.: Harmonicity of unit vector fields with respect to Riemannian natural metrics. Diff. Geom. Appl. 27, 157–169 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Abbassi M.T.K., Calvaruso G., Perrone D.: Some examples of harmonic maps for g-natural metrics. Ann. Math. Blaise Pascal 16, 305–320 (2009)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Abbassi M.T.K., Calvaruso G., Perrone D.: Harmonic maps defined by the geodesic flow. Houst. J. Math. 36(1), 69–90 (2010)MathSciNetGoogle Scholar
  10. 10.
    Abbassi, M.T.K., Calvaruso, G., Perrone, D.: Harmonic sections of tangent bundles equipped with Riemannian g-natural metrics. Q. J. Math. (to appear)Google Scholar
  11. 11.
    Baird P.: Harmonic Maps with Symmetry, Harmonic Morphisms and Deformations of the Metrics. Research Notes in Mathematics. Pitman, Boston (1987)Google Scholar
  12. 12.
    Baird P., Gudmundsson S.: p-Harmonic maps and minimal submanifolds. Math. Ann. 294, 611–624 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Baird, P., Eells, J.: A conservation law for harmonic maps. In: Geometry Symposium, Utrecht 1980. Lecture Notes in Mathematics, vol. 894, pp. 1–25. Springer, Berlin (1981)Google Scholar
  14. 14.
    Baird P., Wood J.C.: Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monograph (NS) 29. Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  15. 15.
    Benyounes M., Lobeau E., Wood C.M.: Harmonic sections of Riemannian vector bundles, and metrics of Cheeger–Gromoll type. Diff. Geom. Appl. 25, 322–334 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Calvaruso G., Perrone D.: Homogeneous and H-contact unit tangent sphere bundles. Aust. J. Math. 88, 323–337 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Cheeger J., Gromoll D.: On the structure of complete manifolds of non negative curvature. Ann. Math. 96, 413–443 (1972)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Chen B.Y., Nagano T.: Harmonic metrics, harmonic tensors and Gauss maps. J. Math. Soc. Jpn. 36(2), 295–313 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Eells J., Sampson J.H.: Harmonic mappings of Riemannian manifolds. Am. Math. J. 86, 109–160 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fuglede B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier (Grenoble) 28, 107–144 (1978)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Gudmundsson, S.: The bibliography of harmonic morphisms.
  22. 22.
    Gudmundsson S., Kappos E.: On the geometry of the tangent bundle with the Cheegerf́bGromoll metric. Tokyo J. Math. 25(1), 75–83 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ishihara T.: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19, 215–229 (1979)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry. Springer-Verlag, Berlin (1993)zbMATHGoogle Scholar
  25. 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)MathSciNetGoogle Scholar
  26. 26.
    Oniciuc C.: The tangent sphere bundles and harmonicity. Analele Stiint. Univ. “Al. I. Cuza” 43, 151–172 (1997)MathSciNetGoogle Scholar
  27. 27.
    Oproiu V.: Some new geometric structures on the tangent bundle. Publ. Math. Debrecen 55, 261–281 (1999)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Perrone D.: Unit vector fields on real space forms which are harmonic maps. Pac. J. Math. 239(1), 89–104 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Perrone D.: Minimality, harmonicity and CR geometry for Reeb vector fields. Int. J. Math. 9, 1189–1218 (2010)CrossRefGoogle Scholar
  30. 30.
    Wood C.M.: An existence theorem for harmonic sections. Manuscr. Math. 68, 69–75 (1990)zbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica “E. De Giorgi”Università del SalentoLecceItaly

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