Hypersurfaces With Parallel M(?)bius Form And Constant Para-Blaschke Eigenvalues In The Unit Sphere

Changping Wang [27] established the M(?)bius geometry of umbilic-free subman-ifolds in the unit sphere, which includes associated with an n-dimensional umbilic-free submanifold x:Mnâ†' Sn+P the introduction of the M(?)bius metric g, the M(?)bius second fundamental form B, the Blaschke tensor A and the M(?)bius form Φ.For λ € R, we call D(λ)= A+λB para-Blaschke tensor, it is also a M(?)bius invariant. Obviously, Blaschke tensor is the special case of para-Blaschke tensor.In [10], the authors proved that for an umbilic-free hypersurface, if its M(?)bius principal curvatures are constants, then the hypersurface is of vanishing M(?)bius form if and only if its M(?)bius form is parallel with respect to the Levi-Civita connection of its M(?)bius metric. In this paper, we consider the condition of para-Blaschke tensor. We show that an umbilic-free hypersurface of the unit sphere is of vanishing M(?)bius form if and only if its M(?)bius form is parallel with respect to the Levi-Civita con-nection of its M(?)bius metric under the condition of having constant para-Blaschke eigenvalues. Moreover, we give a classification of hypersurfaces with paraller M(?)bius form and constant para-Blaschke eigenvalues in Sn+1.