Berger-type Deformed Sasaki Metric and Harmonicity on Tangent Bundles

Year 2022, Volume: 4 Issue: 1, 1 - 16, 24.07.2022

Abderrahım Zagane

Abstract

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the Berger-type deformed Sasaki metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the Berger-type deformed Sasaki metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along a map between Riemannian manifolds, the target manifold being anti-paraKähler equipped with a Berger-type deformed Sasaki metric on its tangent bundle. Also, we 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 anti-paraKähler whose tangent bundle is endowed with a Berger-type deformed Sasaki metric. After that, we study the harmonicity of the identity map on the tangent bundle equipped with the Berger-type deformed Sasaki metric. Finally, we introduce the $\phi$-unit tangent bundle and we also study the harmonicity of the projection map of the $\phi$-unit tangent bundle.

Keywords

Tangent bundles, Berger-type deformed Sasaki metric, harmonic maps, $\varphi$-unit tangent bundle

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