On Riemannian four-manifolds and their twistor spaces: a moving frame approach
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866914946763194368 |
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| author | Catino, Giovanni Dameno, Davide Mastrolia, Paolo |
| author_facet | Catino, Giovanni Dameno, Davide Mastrolia, Paolo |
| contents | In this paper we study the twistor space $Z$ of an oriented Riemannian four-manifold $M$ using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of $Z$ forces the underlying manifold $M$ to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_00323 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Riemannian four-manifolds and their twistor spaces: a moving frame approach Catino, Giovanni Dameno, Davide Mastrolia, Paolo Differential Geometry In this paper we study the twistor space $Z$ of an oriented Riemannian four-manifold $M$ using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of $Z$ forces the underlying manifold $M$ to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold. |
| title | On Riemannian four-manifolds and their twistor spaces: a moving frame approach |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2010.00323 |