Conformal flatness, cohomogeneity one and hypersurfaces of revolution (Q1295469)

A hypersurface \(M \subset \mathbb R^{n+1}\) is called a hypersurface of revolution if it is invariant under the subgroup \(SO_n \subset SO_{n+1}\) which preserves a line. Such a hypersurface is conformally flat and it is a cohomogeneity one manifold, i.e., it admits an isometry group with a codimension one orbit. The authors prove that any conformally flat cohomogeneity one hypersurface \(M\) which is not a hypersurface of revolution can be described as follows. Let \(x =(x,0) , y= (0,y) \) be unit vectors in \(\mathbb R^{n+1} = \mathbb R^2 \oplus \mathbb R^{n-1}\), and \(S^1\) the circle in the plane \(\langle x,y\rangle\) centered at \(cx\) with radius \(r < c\). Then, the hypersurface \(M = (SO_2 \times SO_{n-1})S^1 \).