Generic conformally flat hypersurfaces in ${\mathbb{R}}^4$

Abstract

In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space ${\mathbb{R}}^4$ using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed Möbius form under the Möbius transformation group of ${\mathbb{R}}^4$. Such examples come from cones, cylinders, or rotational hypersurfaces over the surfaces with constant Gaussian curvature in $3$-spheres, Euclidean $3$-spaces, or hyperbolic $3$-spaces, respectively. Second, we investigate the global behavior of the generic conformally flat hypersurface and give some integral formulas about these hypersurfaces.

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