Acta Mathematica Scientia, 2019, 39B(4): 1081-1088 https://doi.org/10.1007/sl0473-019-0412-2 ?Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2019数学物理学报http: actams.wiprn.ac.cn〃 VECTOR FIELDS FOR CONTACT PAIRS*Yue HE (何跃)Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University,
Nanjing 210023, ChinaE-mail: heyue@njnu. edu. cnHai-Long HER (赫海龙)tDepartment of Mathematics, Jinan University, Guangzhou 510632, ChinaE-mail: hailongher@jnu. edu. cnAbstract Let M be a (2k + 22 + 2)-dimensional smooth manifold. For such M、Bande and
Hadj ar introduce a new geome trie 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.Key words Contact geometry; Reeb vector field; contact pair2010 MR Subject Classification 53D10; 57R171 IntroductionThe study of differential forms of constant class was initialed by E. Cartan [7]. Contact (and symplectic) forms are special cases of maximal class. It is well known that there is no local invariant for contact (and symplectic) geometry. For instance, Darboux's theorem for contact manifolds says that every contact structure is locally diffeomorphic to the standard
contact structuTe on R2n+1 given by the standard contact 1-form = dz —刀 Vi^xi (see
[1, 9, 15]). Note that for general contact manifolds, if the contact structure is co-oriented, then it can be defined by a global 1-form. This is also what we consider for the non-maximal class case here. In [3], Bande and H adj ar introduce a new geometrie structure called contact pair. Roughly speaking, a contact pair is a couple of 1-forms of constant classes with complementary kernels and foliations. Donaldson's inspiring work on four-manifolds [8] implies that a pair of differential forms on a manifold likely bring non-trivial geometric structures.Another geometrie object on even dimensional smooth manifolds is symplectic pair which arises from the study of products of harmonic forms and of cohomology of symplectomorphism groups introduced by Kotschick et al (see [4, 12, 13]). Given a 2n-dimensional smooth manifold* Received January 12, 201& revised July 5, 2018. Y. He was supported by the National Natural Science Foundation of China (11671209, 11871278). H.-L. Her was supported by the National Natural Science Foundation of China (11671209) and by the Starting Foundation for Research of Jinan University.Corresponding authori=ln
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