Harmonic maps on the tangent bundle according to the ciconia metric

Year 2025, Volume: 54 Issue: 1, 75 - 89, 28.02.2025

Nour El Houda Djaa , Lokman Bilen , Aydın Gezer

https://doi.org/10.15672/hujms.1343052

Abstract

The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection $\pi :\left( TM,\tilde{g}\right) \rightarrow \left( M_{2n},J,g\right) $, where $\left( M_{2n},J,g\right) $ represents an anti-paraK\"{a}hler manifold and $\left( TM,\tilde{g}\right) $ its tangent bundle with the ciconia metric. Additionally, we study the harmonicity of a vector field $\xi$, treated as mappings from $M$ to $TM$ . In this context, we consider the harmonicity relations between the ciconia metric $\tilde{g}$ and the Sasaki metric $^{S}g$, examining their mutual interactions. Furthermore, we investigate the Schoutan-Van Kampen connection and the Vr\~{a}nceanu connection, both associated with the Levi-Civita connection of the ciconia metric. Our analysis also includes the computation of the mean connections for the Schoutan-Van Kampen and Vr\~{a}nceanu connections, thereby providing insights into their properties. Finally, our exploration extends to the second fundamental form of the identity mapping from $\left( TM,\tilde{g}\right) $ to $\left(TM,\overline{\nabla }^{m}\right) ~$ and $\left( TM,\widetilde{\nabla }^{\ast m}\right) $. Here $\overline{\nabla }^{m}$ and $\widetilde{\nabla }^{\ast m}$ denote the mean connections associated with the Schoutan-Van Kampen and Vr\~{a}nceanu connections, respectively.

Keywords

tangent bundle, ciconia metric, harmonic maps

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