Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric

Authors : Abderrahım ZAGANE, Nour El Houda DJAA

Pages : 83-95

Doi:10.36890/iejg.1033998

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Publication Date : 2022-04-30

Article Type : Research Paper

Abstract :In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the vertical rescaled metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along with a map between Riemannian manifolds, the target manifold is equipped with a vertical rescaled metric on its tangent bundle. Next we also discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, the source manifold being whose tangent bundle is endowed with a vertical rescaled metric. Finally, we study the harmonicity of the tangent map also the harmonicity of the identity map of the tangent bundle.
Keywords : Tangent bundle, horizontal lift, vertical lift, vertical Rescaled metric, harmonic maps