Sasaki structures on general contact manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913162883760128 |
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| author | Grabowska, Katarzyna Grabowski, Janusz Mohseni, Rouzbeh |
| author_facet | Grabowska, Katarzyna Grabowski, Janusz Mohseni, Rouzbeh |
| contents | We extend the notion of a Sasakian structure from the classical setting of a cooriented contact manifold, where it is given by a compatibility between a contact form $η$ and a Riemannian metric $g_M$ on $M$, to the case of an arbitrary contact structure understood as a contact distribution. In the cooriented case, this compatibility can be equivalently expressed by the fact that the symplectic form $ω=\mathrm{d}(s^2η)$ and the cone metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define a Kähler structure on the cone $\mathcal{M}=M\times\mathbb{R}_+$.
Since general contact structures admit canonical realizations as homogeneous symplectic structures $ω$ on principal $\mathbb{R}^\times$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous Kähler structures on $P$. We characterize homogeneous Kähler structures on symplectizations $(P,ω)$ associated with arbitrary contact structures on $M$, and show that they canonically determine a two-sheeted covering $\tilde M$ of $M$ equipped with a contact form. This reduces the problem to the cooriented case and leads to a notion of a generalized Sasakian structure on $M$ associated with a homogeneous Kähler structure on $(P,ω)$.
Moreover, since products of Kähler manifolds are again Kähler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sasaki structures on general contact manifolds Grabowska, Katarzyna Grabowski, Janusz Mohseni, Rouzbeh Differential Geometry Mathematical Physics Symplectic Geometry 53C25, 53D10, 53D05, 32V05, 53D35 We extend the notion of a Sasakian structure from the classical setting of a cooriented contact manifold, where it is given by a compatibility between a contact form $η$ and a Riemannian metric $g_M$ on $M$, to the case of an arbitrary contact structure understood as a contact distribution. In the cooriented case, this compatibility can be equivalently expressed by the fact that the symplectic form $ω=\mathrm{d}(s^2η)$ and the cone metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define a Kähler structure on the cone $\mathcal{M}=M\times\mathbb{R}_+$. Since general contact structures admit canonical realizations as homogeneous symplectic structures $ω$ on principal $\mathbb{R}^\times$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous Kähler structures on $P$. We characterize homogeneous Kähler structures on symplectizations $(P,ω)$ associated with arbitrary contact structures on $M$, and show that they canonically determine a two-sheeted covering $\tilde M$ of $M$ equipped with a contact form. This reduces the problem to the cooriented case and leads to a notion of a generalized Sasakian structure on $M$ associated with a homogeneous Kähler structure on $(P,ω)$. Moreover, since products of Kähler manifolds are again Kähler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices. |
| title | Sasaki structures on general contact manifolds |
| topic | Differential Geometry Mathematical Physics Symplectic Geometry 53C25, 53D10, 53D05, 32V05, 53D35 |
| url | https://arxiv.org/abs/2412.16697 |