HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Abstract

Let (M, g) be a Riemannian manifold. When M is compact and the tangent bundle T M is equipped with the Sasaki metric g s , the only vector fields which define harmonic maps from (M, g) to (T M, g s ), are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on T M , are particular examples of g-natural metrics. We equip T M with an arbitrary Riemannian g-natural metric G, and investigate the harmonicity of a vector field V of M , thought as a map from (M, g) to (T M, G). We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.

Unit vector fields on real space forms which are harmonic maps

“…However, there is no general existence theory of harmonic mappings if the target manifold does not satisfy the nonpositivity curvature condition. This fact makes it interesting to find harmonic maps defined by vector fields and unit vector fields [Abbassi et al 2007;2008;Benyounes et al 2007b;2007a;Ishihara 1979;Nouhaud 1977;Perrone 2003;2005;Rukimbira 2002;Tsukada and Vanhecke 2001].…”

Instability of the geodesic flow for the energy functional

“…These metrics depend on several smooth functions from ‫ޒ‬ C D OE0; C1/ to ‫ޒ‬ and as their name suggests, they arise from a very "natural" construction starting from a Riemannian metric g over M (see [Abbassi and Sarih 2005;Abbassi et al 2010a] and the references in [Abbassi 2008]). Given an arbitrary g-natural metric G on the tangent bundle TM of a Riemannian manifold .M; g/, there are six smooth functions˛i,ˇi W ‫ޒ‬ C !…”

Harmonic morphisms and Riemannian geometry of tangent bundles

“…Harmonic sections of T M and T 1 M, equipped with Riemannian g-natural metrics, were studied in [7,8,10,28,29], while the harmonicity of the geodesic flow was investigated in [9]. Up to our knowledge, few results have been obtained in so far about the harmonicity of the canonical projections.…”