The Global Property of Generic Conformally Flat Hypersurfaces in R4

Abstract

A conformally flat hypersurface f:M3→R4 in the four-dimensional Euclidean space R4 is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in R4 using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of R4. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces.