Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866929427998310400 |
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| author | Nejadahmad, Masumeh Moghaddam, Hamid Reza Salimi |
| author_facet | Nejadahmad, Masumeh Moghaddam, Hamid Reza Salimi |
| contents | In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(α,β)$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ and give necessary and sufficient conditions for them to be of Douglas type. Then, the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_00685 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups Nejadahmad, Masumeh Moghaddam, Hamid Reza Salimi Differential Geometry 53B21, 22E60, 22E15 In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(α,β)$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ and give necessary and sufficient conditions for them to be of Douglas type. Then, the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given. |
| title | Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups |
| topic | Differential Geometry 53B21, 22E60, 22E15 |
| url | https://arxiv.org/abs/1903.00685 |