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 $(T{𝕊}^2,G)$.

Classification : 58E20, 53C43
Mots clés : harmonic map, tangent bundle, vector fields, $g$-natural metrics, spaces of constant curvature.

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