Finite Type Pseudo-umbilical Submanifolds in a Hypersphere

where Xo is a constant vector and AX,-t — XitX{t, t = 1, 2, . . . , k. If in particular all eigenvalues {Atl, . . . , A,t} are mutually different, then M is said to be of i-type. A Jfc-type submanifold is said to be null if one of the A;t, t — 1, 2, . . . , k, is null. It is easy to see that if M is compact, then Xo in (1.1) is exactly the centre of mass in E . A submanifold M of a hypersphere S~ of E is called mass-symmetric in 5 m 1 if the centre of mass of M in E is the centre of the hypersphere S~ in E (see [1] for details). In terms of finite type submanifolds, a well-known result of Takahashi [6] says that a submanifold M in E is of 1-type if and only if it is either a minimal submanifold of E or a minimal submanifold of a hypersphere of E. In the first case M is of null 1-type and in the last case M is mass-symmetric in S~. In [3] it was proved that a compact 2-type hypersurface of a hypersphere is mass-symmetric if and only if it has constant mean curvature. In [5] it was proved that every 2-type pseudo-umbilical submanifold of E with constant mean curvature is either spherical or null 2-type. And in [4] some examples were given for spherical 2-type pseudo-umbilical surfaces, which is mass-symmetric. In this article we prove the following.