Some examples of harmonic maps for $g$-natural metrics

Abbassi, Mohamed Tahar Kadaoui; Calvaruso, Giovanni; Perrone, Domenico

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 305-320.

On produit des nouveaux exemples d’applications harmoniques, ayant chacune comme espace de départ une variété $(M,g)$ á courbure constante et comme espace d’arrivée son fibré tangent $TM$, muni d’une métrique $g$-naturelle Riemannienne appropriée. En particulier, on va déterminer une famille de métriques $g$-naturelles Riemanniennes $G$ sur $T{𝕊}^2$, par rapport auxquelles tous les champs de vecteurs gradients conformes définissent des applications harmoniques de ${𝕊}^2$ dans $(T{𝕊}^2,G)$.

We produce new examples of harmonic maps, having as source manifold a space $(M,g)$ of constant curvature and as target manifold its tangent bundle $TM$, equipped with a suitable Riemannian $g$-natural metric. In particular, we determine a family of Riemannian $g$-natural metrics $G$ on $T{𝕊}^2$, with respect to which all conformal gradient vector fields define harmonic maps from ${𝕊}^2$ into $(T{𝕊}^2,G)$.

[1] Abbassi, M. T. K.; Calvaruso, G.; Perrone, D. Harmonic sections of tangent bundles equipped with Riemannian $g$-natural metrics (2007) (Submitted, online version: arXiv:math/0710.3668) | MR | Zbl

[2] Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno), Volume 41 (2005) no. 1, pp. 71-92 | EuDML | MR | Zbl

[3] Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti On some hereditary properties of Riemannian $g$-natural metrics on tangent bundles of Riemannian manifolds, Differential Geom. Appl., Volume 22 (2005) no. 1, pp. 19-47 | DOI | MR | Zbl

[4] Baird, Paul; Wood, John C. Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, 29, The Clarendon Press Oxford University Press, Oxford, 2003 | MR | Zbl

[5] Benyounes, M.; Loubeau, E.; Wood, C. M. Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type, Differential Geom. Appl., Volume 25 (2007) no. 3, pp. 322-334 | DOI | MR | Zbl

[6] Benyounes, M.; Loubeau, E.; Wood, C.M. Harmonic maps and sections on spheres (2008) (preprint)

[7] Eells, J.; Lemaire, L. A report on harmonic maps, Bull. London Math. Soc., Volume 10 (1978) no. 1, pp. 1-68 | DOI | MR | Zbl

[8] Eells, James Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160 | DOI | MR | Zbl

[9] Ishihara, Tôru Harmonic sections of tangent bundles, J. Math. Tokushima Univ., Volume 13 (1979), pp. 23-27 | MR | Zbl

[10] Kolář, Ivan; Michor, Peter W.; Slovák, Jan Natural operations in differential geometry, Springer-Verlag, Berlin, 1993 | MR | Zbl

[11] Nouhaud, Odette Applications harmoniques d’une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris Sér. A-B, Volume 284 (1977) no. 14, p. A815-A818 | MR | Zbl

[12] Oniciuc, C. The tangent bundles and harmonicity, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), Volume 43 (1997) no. 1, p. 151-172 (1998) | MR | Zbl

[13] Sampson, J. H. Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4), Volume 11 (1978) no. 2, pp. 211-228 | Numdam | MR | Zbl

[14] Sanini, Aristide Maps between Riemannian manifolds with critical energy with respect to deformations of metrics, Rend. Mat. (7), Volume 3 (1983) no. 1, pp. 53-63 | MR | Zbl

[15] Urakawa, Hajime Calculus of variations and harmonic maps, Translations of Mathematical Monographs, 132, American Mathematical Society, Providence, RI, 1993 (Translated from the 1990 Japanese original by the author) | MR | Zbl

[16] Xin, Y. L. Some results on stable harmonic maps, Duke Math. J., Volume 47 (1980) no. 3, pp. 609-613 | DOI | MR | Zbl