Conformally flat hypersurfaces and Guichard nets

Abstract

100 years ago Cartan first posed the question to find a geometric characterization of 3-dimensional conformally flat hypersurfaces - this thesis aims to make further steps in this direction. We introduce the novel notion of a Suyama-metric and show its existence in the conformal class of these hypersurfaces. Due to a classification result by Hertrich-Jeromin, generic conformally flat hypersurfaces can be equivalently examined from the point of view of Guichard nets, particular curvature line coordinates viewed as triply orthogonal systems in R 3. In this light we present associated systems for Guichard nets and characterize Guichard nets consisting of isothermic coordinate surfaces. To obtain the associated systems, we geometrically characterize the coordinate surfaces of a Guichard net as G-surfaces, a generalization of isothermic and Guichard surfaces. Suitably chosen associated surfaces of the G-surfaces of the Guichard net, then form a particular Combescure related triply orthogonal system - the aforementioned associated system. A class of Guichard nets that is highlighted in this thesis is that of isothermic Guichard nets. We give a characterization in terms of their metrics and relate them to a classification obtained by Darboux. Those containing one family of totally umbilic coordinate surfaces are investigated in detail, emphasising the interplay between these Guichard nets and the geometry of the corresponding conformally flat hypersurfaces. The observation that one class of isothermic Guichard nets is obtained by "inverting" elliptic coordinates reveals an interesting relation to the problem of R-separability of Laplace's equation. This motivates the study of inverted triply orthogonal systems, which provide explicit examples of isothermic Guichard nets. We prove that the existence of a Suyama-metric in the conformal class of a metric satisfying the Guichard condition detects conformal flatness. This result, viewed in the realm of triply orthogonal systems, provides a refinement of a potential found by Darboux. To gain more geometric insight, we consider the situation for particular classes of Guichard nets and determine the Suyama-metrics for the isothermic Guichard nets. Moreover, we extend the theory of O-surfaces to 3-dimensional hypersurfaces, a concept inspired by integrable geometry and introduced by Konopelchenko and Schief. Conformally flat 3-dimensional submanifolds are characterized as O-hypersurfaces using their Guichard dual.