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

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

Annales mathématiques Blaise Pascal, Tome 16 (2009), p. 305-320 / Harvested from Numdam

Résumé

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 $T\mathbb{S}^2$, par rapport auxquelles tous les champs de vecteurs gradients conformes définissent des applications harmoniques de $\mathbb{S}^2$ dans $(T\mathbb{S}^2, G)$.

Abstract

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 $T\mathbb{S}^2$, with respect to which all conformal gradient vector fields define harmonic maps from $\mathbb{S}^2$ into $(T\mathbb{S}^2,G)$.

Détail

Publié le : 2009-01-01

MR 2568868 | Zbl 1183.58008

DOI : https://doi.org/10.5802/ambp.269
Classification:  58E20,  53C43

Bibliographie

[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 2805204 | Zbl 1226.53062

[2] Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno), Tome 41 (2005) no. 1, pp. 71-92 | MR 2142144 | Zbl 1114.53015

[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., Tome 22 (2005) no. 1, pp. 19-47 | Article | MR 2106375 | Zbl 1068.53016

[4] Baird, Paul; Wood, John C. Harmonic morphisms between Riemannian manifolds, The Clarendon Press Oxford University Press, Oxford, London Mathematical Society Monographs. New Series, Tome 29 (2003) | MR 2044031 | Zbl 1055.53049

[5] Benyounes, M.; Loubeau, E.; Wood, C. M. Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type, Differential Geom. Appl., Tome 25 (2007) no. 3, pp. 322-334 | Article | MR 2330461 | Zbl 1128.53037

[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., Tome 10 (1978) no. 1, pp. 1-68 | Article | MR 495450 | Zbl 0401.58003

[8] Eells, James Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Tome 86 (1964), pp. 109-160 | Article | MR 164306 | Zbl 0122.40102

[9] Ishihara, Tôru Harmonic sections of tangent bundles, J. Math. Tokushima Univ., Tome 13 (1979), pp. 23-27 | MR 563393 | Zbl 0427.53019

[10] Kolář, Ivan; Michor, Peter W.; Slovák, Jan Natural operations in differential geometry, Springer-Verlag, Berlin (1993) | MR 1202431 | Zbl 0782.53013

[11] Nouhaud, Odette Applications harmoniques d’une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris Sér. A-B, Tome 284 (1977) no. 14, p. A815-A818 | MR 431035 | Zbl 0349.53015

[12] Oniciuc, C. The tangent bundles and harmonicity, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), Tome 43 (1997) no. 1, p. 151-172 (1998) | MR 1679113 | Zbl 0989.53039

[13] Sampson, J. H. Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4), Tome 11 (1978) no. 2, pp. 211-228 | Numdam | MR 510549 | Zbl 0392.31009

[14] Sanini, Aristide Maps between Riemannian manifolds with critical energy with respect to deformations of metrics, Rend. Mat. (7), Tome 3 (1983) no. 1, pp. 53-63 | MR 710809 | Zbl 0514.53037

[15] Urakawa, Hajime Calculus of variations and harmonic maps, American Mathematical Society, Providence, RI, Translations of Mathematical Monographs, Tome 132 (1993) (Translated from the 1990 Japanese original by the author) | MR 1252178 | Zbl 0799.58001

[16] Xin, Y. L. Some results on stable harmonic maps, Duke Math. J., Tome 47 (1980) no. 3, pp. 609-613 | Article | MR 587168 | Zbl 0513.58019