| Volume-preserving Fields and Reeb Fields on 3-manifolds |
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| 引用本文： | Hong Jun LI. Volume-preserving Fields and Reeb Fields on 3-manifolds[J]. 数学学报(英文版), 2006, 22(4): 971-988. DOI: 10.1007/s10114-005-0543-3 |
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| 作者姓名： | Hong Jun LI |
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| 作者单位： | Science College, Xi＇an Jiaotong University, Xi＇an 710049, P. R. China |
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| 基金项目： | Acknowlegements I wish to express my gratitude to Xiaosong Lin, who encouragement, many useful discussions, patient guidance and correction for th I am also greatly aided through conversations with Haibao Duan and Banghe is in fact Banghe Li＇s observation. gives me great e present paper. Li. Lemma 4.8 |
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| 摘 要： | Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.
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| 关 键 词： | 体积保存场 接触模板 泊松矩阵 并行化 Reeb场 |
| 收稿时间： | 2003-03-10 |
| 修稿时间： | 2003-03-102004-06-18 |
Volume–preserving Fields and Reeb Fields on 3–manifolds |
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| Hong Jun Li. Volume–preserving Fields and Reeb Fields on 3–manifolds[J]. Acta Mathematica Sinica(English Series), 2006, 22(4): 971-988. DOI: 10.1007/s10114-005-0543-3 |
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| Authors: | Hong Jun Li |
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| Affiliation: | (1) Science College, Xi’an Jiaotong University, Xi’an 710049, P. R. China |
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| Abstract: | Volume–preserving field X on a 3–manifold is the one that satisfies L X Ω ≡ 0 for some volume Ω. The Reeb vector field of a contact form is of volume–preserving, but not conversely. On the basis of Geiges–Gonzalo’s parallelization results, we obtain a volume–preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume–preserving fields. From many aspects, we discuss the distinction between volume–preserving fields and Reeb–like fields. We establish a duality between volume–preserving fields and h–closed 2–forms to understand such distinction. We also give two kinds of non–Reeb–like but volume–preserving vector fields to display such distinction. |
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| Keywords: | Volume-preserving fields Contact forms Reeb-like fields Volume-preserving sphere Poisson matrices |
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