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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.10591 |
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| _version_ | 1866913473473019904 |
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| author | Dong, Yuxin Ren, Yibin |
| author_facet | Dong, Yuxin Ren, Yibin |
| contents | In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form θ, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{θ}. This induces a natural Riemannian metric g_{h,θ} on M compatible with the structure. The synthetic object (M,θ,J,h) will be called a pseudo-Hermitian manifold, which generalizes the usual notion of pseudo-Hermitian manifold (M,θ,J,L_{θ}) in the literature. Our purpose is to investigate the differential-geometric aspect of pseudo-Hermitian manifolds. By imitating Hermitian geometry, we find a canonical connection on (M,θ,J,h), which generalizes the Tanaka-Webster connection on (M,θ,J,L_{θ}). We define the pseudo-Kähler 2-form by g_{h,θ} and J; and introduce the notion of a pseudo-Kähler manifold, which is an analogue of a Kähler manifold. It turns out that (M,θ,J,L_{θ}) is pseudo-Kählerian. Using the structure equations of the canonical connection, we derive some curvature and torsion properties of a pseudo-Hermitian manifold, in particular of a pseudo-Kähler manifold. Then some known results in Riemannian geometry are generalized to the pseudo-Hermitian case. These results include some Cartan type results. As an application, we give a new proof for the classification of Sasakian space forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10591 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On differential geometry of non-degenerate CR manifolds Dong, Yuxin Ren, Yibin Differential Geometry In this paper, we consider a non-degenerate CR manifold (M,H(M),J) with a given pseudo-Hermitian 1-form θ, and endow the CR distribution H(M) with any Hermitian metric h instead of the Levi form L_{θ}. This induces a natural Riemannian metric g_{h,θ} on M compatible with the structure. The synthetic object (M,θ,J,h) will be called a pseudo-Hermitian manifold, which generalizes the usual notion of pseudo-Hermitian manifold (M,θ,J,L_{θ}) in the literature. Our purpose is to investigate the differential-geometric aspect of pseudo-Hermitian manifolds. By imitating Hermitian geometry, we find a canonical connection on (M,θ,J,h), which generalizes the Tanaka-Webster connection on (M,θ,J,L_{θ}). We define the pseudo-Kähler 2-form by g_{h,θ} and J; and introduce the notion of a pseudo-Kähler manifold, which is an analogue of a Kähler manifold. It turns out that (M,θ,J,L_{θ}) is pseudo-Kählerian. Using the structure equations of the canonical connection, we derive some curvature and torsion properties of a pseudo-Hermitian manifold, in particular of a pseudo-Kähler manifold. Then some known results in Riemannian geometry are generalized to the pseudo-Hermitian case. These results include some Cartan type results. As an application, we give a new proof for the classification of Sasakian space forms. |
| title | On differential geometry of non-degenerate CR manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2408.10591 |