On a metric symplectization of a contact metric manifold
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913439440437248 |
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| author | Alape, Sannidhi |
| author_facet | Alape, Sannidhi |
| contents | In this article, we investigate metric structures on the symplectization of a contact metric manifold and prove that there is a unique metric structure, which we call the metric symplectization, for which each slice of the symplectization has a natural induced contact metric structure. We then study the curvature properties of this metric structure and use it to establish equivalent formulations of the $(κ, μ)$-nullity condition in terms of the metric symplectization. We also prove that isomorphisms of the metric symplectizations of $(κ, μ)$-manifolds determine $(κ, μ)$-manifolds up to D-homothetic transformations. These classification results show that the metric symplectization provides a unified framework to classify Sasakian manifolds, K-contact manifolds and $(κ, μ)$-manifolds in terms of their symplectizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a metric symplectization of a contact metric manifold Alape, Sannidhi Differential Geometry Symplectic Geometry 53C15, 53C25, 53D10 In this article, we investigate metric structures on the symplectization of a contact metric manifold and prove that there is a unique metric structure, which we call the metric symplectization, for which each slice of the symplectization has a natural induced contact metric structure. We then study the curvature properties of this metric structure and use it to establish equivalent formulations of the $(κ, μ)$-nullity condition in terms of the metric symplectization. We also prove that isomorphisms of the metric symplectizations of $(κ, μ)$-manifolds determine $(κ, μ)$-manifolds up to D-homothetic transformations. These classification results show that the metric symplectization provides a unified framework to classify Sasakian manifolds, K-contact manifolds and $(κ, μ)$-manifolds in terms of their symplectizations. |
| title | On a metric symplectization of a contact metric manifold |
| topic | Differential Geometry Symplectic Geometry 53C15, 53C25, 53D10 |
| url | https://arxiv.org/abs/2407.15057 |