Browsing by Browse by FOR 2008 "010106 Lie Groups, Harmonic and Fourier Analysis"

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.

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call FRT-metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of 3-dimensional CR-structures in terms of the Ricci curvature of the FRT-metrics in the spirit of the results by Lewandowski et al. in [37] and also Hill et al. in [25]. We also study higher dimensional versions of shearfree null congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure.

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R²+ x R²-. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.