Classification of hypersurfaces with constant Möbius curvature in \({\mathbb{S}^{m+1}}\)
Let \(x:M \rightarrow S^{m+1}\) be an \(m\)-dimensional umbilic-free hypersurface \(M\) immerse in \(S^{m+1}\), with the standard metric \(I=dx \cdot dx\). By using the positive function \(\rho=\sqrt{\frac{m}{m-1} \| II - \frac{1}{m} tr(II)I \|}\), where \(II\) is the second fundamental form, the authors define the \((0,2)\) tensor \(g=\rho^2 I\) called Möbius metric, which is invariant under conformal transformations of \(S^{m+1}\). The curvature induced by the metric \(g\) is known as Möbius curvature. The principal result in this paper is the classification of \(m\)-dimensional hypersurfaces \(M\) in \(S^{m+1}\) with constant Möbius sectional curvature by using the Möbius geometry method of submanifolds established by \textit{C. P. Wang} [Manuscr. Math. 96, No. 4, 517--534 (1998; Zbl 0912.53012)].
scientific article; zbMATH DE number 6050613
scientific article