Immersed hypersurfaces in the unit sphere \(S^{m+1}\) with constant Blaschke eigenvalues
Let \(x:M^m \rightarrow S^{m+p}\) be an immersion into the sphere \(S^{m+p}\) without umbilic points. One can introduce the following Möbius invariants: The Möbius metric \(g,\) the Möbius form \(\phi,\) the Blaschke tensor \(A,\) and the Möbius second fundamental form \(B.\) The eigenvalues of \(B\) are called \textit{Möbius principal curvatures,} the eigenvalues of \(A\) are called \textit{Blaschke eigenvalues.} In this paper a classification of hypersurfaces \(x:M^{m}\rightarrow S^{m+1}\) with vanishing Möbius form and two constant and distinct Blaschke eigenvalues is presented. Another result is the classification of immersed and umbilic free hypersurfaces in \(S^4\) with vanishing Möbius form and constant Blaschke eigenvalues.
Blaschke eigenvalues, Möbius metric, Möbius second fundamental form
scientific article; zbMATH DE number 5162331
scientific article