Normalization of Singular Contact Forms and Primitive 1-forms

IF 0.3 Q4 MATHEMATICS Acta Mathematica Vietnamica Pub Date : 2024-10-09 DOI:10.1007/s40306-024-00545-5
Kai Jiang, Hong Minh Truong, Nguyen Tien Zung
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Abstract

A differential 1-form \(\alpha \) on a manifold of odd dimension \(2n+1\), which satisfies the contact condition \(\alpha \wedge (d\alpha )^n \ne 0\) almost everywhere, but which vanishes at a point O, i.e., \(\alpha (O) = 0\), is called a singular contact form at O. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form \(\omega \) in dimension 2n, i.e., differential 1-forms \(\gamma \) which vanish at a point and such that \(d\gamma = \omega \), and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin (1975), Webster (Amer. J. Math. 109, 807–832 (1987)) and Zhitomirskii (1986, 1992). Besides the classical normalization techniques, such as the step-by step normalization methods based on the cohomological equations and the Moser path method, we also use the toric approach to the normalization problem for dynamical systems (Jiang et al. 2019; Zung, Ann. Math. 161, 141–156 2005; Zung 2016; Zung, Arch. Rational Mech. Anal. 229, 789–833 (2018)).

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奇异接触形式和原始 1-形式的规范化
奇维度流形(2n+1)上的微分 1-form \(\alpha \),几乎在所有地方都满足接触条件 \(\alpha \wedge (d\alpha )^n \ne 0\) ,但是在点 O 上消失,即 \(\alpha (O) = 0\) ,被称为在 O 上的奇异接触形式、\本文的目的是研究这种奇异接触形式的局部法形式(形式的、解析的和光滑的)。我们的研究很自然地引出了维数为 2n 的交点形式 \(\omega \)的奇异基元 1-forms 的法形式研究,即在某点消失且使得 \(d\gamma = \omega \)的微分 1-forms \(\gamma),以及它们相应的共形向量场。我们的结果是对其他作者,特别是 Lychagin (1975)、Webster (Amer. J. Math.) 等人之前结果的扩展和改进。J. Math.109, 807-832 (1987)) 和 Zhitomirskii (1986, 1992)。除了经典的归一化技术,如基于同调方程的分步归一化方法和莫瑟路径法,我们还利用环方法来解决动力系统的归一化问题(Jiang 等,2019;Zung,Ann.Math.161, 141-156 2005; Zung 2016; Zung, Arch.Rational Mech.Anal.229, 789-833 (2018)).
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CiteScore
0.90
自引率
0.00%
发文量
23
期刊介绍: Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.
期刊最新文献
Preface Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations Normalization of Singular Contact Forms and Primitive 1-forms A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian Note on the Linear Independence of Alternating Multiple Zeta Values in Positive Characteristic
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