Normal forms of para-CR hypersurfaces

Abstract

We consider hypersurfaces of finite type in a direct product space ${\mathbb{R}}^2\times {\mathbb{R}}^2$, which are analogues to real hypersurfaces of finite type in ${\mathbb{C}}^2$. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in ${\mathbb{C}}^2$. For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted $\ell$-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.