The duality of conformally flat manifolds

Abstract

In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points (i.e. admissible GCF-manifolds), * frontals as hypersurfaces in the lightcone Q^{n+1}_+, * frontals as hypersurfaces in the hyperbolic space H^{n+1}, * sapacelike frontals as hypersurfaces in the de Sitter space S^{n+1}_1. Recently, the duality of conformally flat Riemannian manifolds was found by several geometers. In our setting, this duality can be explained via the existence of a two-fold map of the congruent classes of admissible GCF-manifolds into that of frontals in H^{n+1}. It should be remarked that the dual conformally flat metric may have degenerate points even when the original conformally flat metric is positive definite. This is the reason why we consider conformally flat manifolds with singular points. In fact, the duality is an involution on the set of admissible GCF-manifolds. The case n=2 requires an exceptional treatment, since any Riemannian 2-manifolds are conformally flat. At the end of this paper, we also determine the moduli space of isometric immersions of a given simply connected Riemannian 2-manifold into the lightcone Q^{3}_+.