Rigidity of Lie affine foliations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916598048096256 |
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| author | Geudens, Stephane |
| author_facet | Geudens, Stephane |
| contents | In previous work by El Kacimi Alaoui-Guasp-Nicolau, a cohomological criterion is given for a Lie $\mathfrak{g}$-foliation on a compact manifold to be rigid among nearby Lie foliations. Our aim is to look for examples of this rigidity statement in case the Lie foliation is modeled on the two-dimensional non-abelian Lie algebra $\mathfrak{g}=\mathfrak{aff}(1)$. We study the relevant cohomology group in detail, showing that it can be expressed in terms of Morse-Novikov cohomology. We find the precise conditions under which it vanishes, which yields many examples of rigid Lie affine foliations. In particular, we show that any Lie affine foliation on a compact, connected, orientable manifold of dimension $3$ or $4$ is rigid when deformed as a Lie foliation. Our results rely on a computation of the Morse-Novikov cohomology groups associated with a nowhere-vanishing closed one-form with discrete period group, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity of Lie affine foliations Geudens, Stephane Differential Geometry 53C12 In previous work by El Kacimi Alaoui-Guasp-Nicolau, a cohomological criterion is given for a Lie $\mathfrak{g}$-foliation on a compact manifold to be rigid among nearby Lie foliations. Our aim is to look for examples of this rigidity statement in case the Lie foliation is modeled on the two-dimensional non-abelian Lie algebra $\mathfrak{g}=\mathfrak{aff}(1)$. We study the relevant cohomology group in detail, showing that it can be expressed in terms of Morse-Novikov cohomology. We find the precise conditions under which it vanishes, which yields many examples of rigid Lie affine foliations. In particular, we show that any Lie affine foliation on a compact, connected, orientable manifold of dimension $3$ or $4$ is rigid when deformed as a Lie foliation. Our results rely on a computation of the Morse-Novikov cohomology groups associated with a nowhere-vanishing closed one-form with discrete period group, which may be of independent interest. |
| title | Rigidity of Lie affine foliations |
| topic | Differential Geometry 53C12 |
| url | https://arxiv.org/abs/2402.04633 |