On the normal forms for Pfaffian systems (Q2372695)

Let \(M\) be a \(C^\infty\)-manifold of dimension \(n\) and \(T^*M\) its cotangent bundle. Let \(C^\infty(M)\) denote the ring of \(C^\infty\)-functions on \(M\) and \(\Gamma(T^*M)\) the \(C^\infty(M)\)-module of global \(C^\infty\)-sections of \(T^*M\). A \(C^\infty(M)\)-submodule of \(\Gamma(T^*M)\) is called a Pfaffian system \(S\) on \(M\). Let \(J^r(\mathbb R^h,\mathbb R^q)=(x_{\alpha_1},z^i,z^1_{\alpha_1},\dots,z^1_{\alpha_1\dots\alpha_r})\) be the jet manifold. There is a canonical Pfaffian system on \(J^r(\mathbb R^h,\mathbb R^q)\), called the contact system, which is the Pfaffian system generated by the 1-forms \(\omega^i = dz^i - \sum_{\alpha_1=1}^hz^i_{\alpha_1}dx_{\alpha_1}\), \(\omega^i_{\alpha_1} = dz^i_{\alpha_1} - \sum_{\alpha_2=1}^hz^i_{\alpha_1\alpha_2}dx_{\alpha_2}\), \(\dots\), \(\omega^i_{\alpha_1\dots\alpha_{r-1}} =dz^i_{\alpha_1\dots\alpha_{r-1}} -\sum_{\alpha_r=1}^hz^1_{\alpha_1\dots\alpha_r}dx_{\alpha_r}\). A contact system restricted to a submanifold of \(J^r(\mathbb R^h,\mathbb R^q)\) is called the Pfaffian system associated to a system of partial differential equations. In this paper, the author studies local normal forms of Pfaffian systems and gives a necessary and sufficient condition for a Pfaffian system to be transformed into the contact system on the jet manifold \(J^r(\mathbb R^h,\mathbb R^q)\) or into the Pfaffian system associated to a system of partial differential equations. The properties of relative polarization are used to prove this result which is the generalization of the Darboux theorem on a Pfaffian equation of constant class.