Normal forms of para-CR hypersurfaces

Abstract

We consider hypersurfaces of finite type in a direct product space R2×R2, which are analogues to real hypersurfaces of finite type in C2. We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in C2. 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 ℓ-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.