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 p...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.展开更多
Let M be a(2 k + 2 l + 2)-dimensional smooth manifold. For such M, Bande and Hadjar introduce a new geometric structure called contact pair which roughly is a couple of 1-forms of constant classes with complementary k...Let M be a(2 k + 2 l + 2)-dimensional smooth manifold. For such M, Bande and Hadjar introduce a new geometric structure called contact pair which roughly is a couple of 1-forms of constant classes with complementary kernels and foliations. We show the relationship between a pair of vector fields for a contact pair and a quadruple of functions on M. This is a generalization of the classical result for contact manifolds.展开更多
文摘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.
基金Supported by the National Natural Scielice Foundation of China under Grant Nos.6057315360533080(国家自然科学基金)+5 种基金the National High-Tech Research and Development Plan of China under Grant No.2006AA01Z314(国家高技术研究发展计划(863))the Program for New Century Excellent Talents in University of China under Grant No.NCET-05-0519(新世纪优秀人才支持计划)the Natural science Foundation of Zhejiang Provilice of China under Grant No.R105431(浙江省自然科学基金)the Open Project Program of the State Key Laboratory of CAD&CGZhejiang UniversityChina under Grant No.A0805(浙江大学CAD&CG国家重点实验宣开放课题)
基金supported by the National Natural Science Foundation of China(11671209,11871278)by the Starting Foundation for Research of Jinan University
文摘Let M be a(2 k + 2 l + 2)-dimensional smooth manifold. For such M, Bande and Hadjar introduce a new geometric structure called contact pair which roughly is a couple of 1-forms of constant classes with complementary kernels and foliations. We show the relationship between a pair of vector fields for a contact pair and a quadruple of functions on M. This is a generalization of the classical result for contact manifolds.
