Submanifolds With Vanishing Moebius Form In The Unit Sphere

Let M^mbe a m-dimensional submanifold in the n-dimensional unit sphere Sⁿ without umbilic point. Four basic invariants of M^m under the Moebius transformation group of Sⁿ are a symmetrice positive definite 2-form g called the Moebius metric , a section of normal bundle B called the Moebius second fundamental form, a 1-form φ called Moebius form and a symmetric (0,2) tensor A called Blaschke tensor.In this paper,let M^m be a m-dimensional submanifold with vanishing MoebiusForm in Sⁿ,denote D = A = A-1/mtrA·id, we calculate the Laplace operator of||B||² ,when ||D|| satisfies some conditions, then we discuss the property of M^m ,wewill prove the following rigidity result:Main Theorem: Let M^m(n=m+p) be a m-dimensional submanifold with vanishingMoebius form in Sⁿ, if 0 ≤ 　||P|| ≤ 2trA - [1 + 1/2sgn(p -1)] ,thenR=const (I) and x(M) is Moebius equivalent to either (i) S¹((1-r²)^1/2)×S^m-1(r)(0m+1, (ii) the image of R¹Ã—S^m-1(1) in R^m+1 unter the project σ,(iii) the image of H1((1+ r²)^1/2)×S^m-1(r) (r>0) in H^m+1 unter the project Ï„, (iv)Versonese surface in S⁴;(â…¡) ||D|| = 0 and x(M) is Moebius equivalent to Clifford minimal toriin S^m+1,1≤k≤m-1...