Normalization of Singular Contact Forms and Primitive 1-forms

Abstract A differential 1-form $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> on a manifold of odd dimension $$2n+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , which satisfies the contact condition $$\alpha \wedge (d\alpha )^n \ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∧</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> almost everywhere, but which vanishes at a point O , i.e., $$\alpha (O) = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>O</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> in dimension 2 n , i.e., differential 1-forms $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> which vanish at a point and such that $$d\gamma = \omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , 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)).