Immersed Hypersurfaces in the Unit Sphere S m+1 with Constant Blaschke Eigenvalues

Abstract

For an immersed submanifold x: Mm → Sn in the unit sphere Sn without umbilics, an eigenvalue of the Blaschke tensor of x is called a Blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant Blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in Sm+1 with vanishing Mobius form and constant Blaschke eigenvalues, in case (1) x has exact two distinct Blaschke eigenvalues, or (2) m = 3. With these classifications, some interesting examples are also presented.