With Parallel Mean Curvature Submanifolds Geometric Rigidity Theorem

In the present thesis, we mainly study the rigidity problems for submanifolds. We prove intrinsic rigidity theorems on terms of sectional curvature, scalar curvature and Ricci curvature for submanifolds with parallel mean curvature and positive curvature in a space form. We get a generalization of the famous rigidity theorems due to S. T. Yau and T. Itoh and we generalize the correlational works for minimal submanifolds due to A. Li , J. Li, N. Ejili and the rigidity theorems for compact submanifolds with parallel mean curvature in a sphere due to Z. Sun. We obtain some best Pinching constants at the same time.In chapter 3, we prove the following main theoremsTheorem 3.1. Let Mⁿ be n-dimensional oriented compact submanifold with parallel mean curvature (H ≠ 0) in F^n+P(c). If K_M > 0, then (i) If p ≤2 then M is a totally umbilical sphere Sn F^n+P(c). (ii)If≥ 3, and ifthen M is a totally umbilical sphereTheorem 3.2. Let Mⁿ be n-dimensional oriented compact submanifold with parallel mean curvature (H≠ 0) in F^n+P(c), where p≥ 2 and c + H² > 0. Ifthen either M is the totally umbilical sphere Sⁿ, the standard immersion of the product of two spheres or the Veronese surface inIn chapter 4, the purpose is to prove rigidity theorems for submanifold with parallel mean curvature, and the ambient space to general Riemannian manifold F^n+P(c), where c is constant. We investigate the rigidity theories of scalar curvature and Ricci curvature . Thus we obtain the generalizations of results of A. M. Li, J. M. Li [8] , N. Ejili [4] and Z. Q. Sun [11]. We prove the followingTheorem 4.1. Let Mn ^-> Fn+P(c) be an isometrically immersion. Mn is an n-dimensional compact pseudo-umbilical submanifold with parallel mean curvature, p > 1, c + H2 > 0. If the scalar curvature of M" is not less than |n(3n - 5)(c + H2), then Mn is totally umbilical or Mn is a Veronese surface in Si(^H2) when n = 2.Theorem 4.2. Let M" <—> Fn+P(c) be an isometrically immersion. M" is an n-dimensional compact pseudo-umbilical submanifold with parallel mean curvature, p > 1, c + H2>0, n > 4. If the Ricri curvature of Mn is larger than (n - 2) (c + H2), then M" is totally umbilical.From Z. Q. Sun '61, we generalize his resultsTheorem 4.3. Let M" c Fn+P(c) be a compact and connected submanifold with parallel mean curvature, p > l,n > 4. If the Ricci curvature of Mn is not less than ^^Eric + H2), then Mn is totally umbilical. When c = 1, it is the result of Z. Q. Sun.What' more, a better result is the followingTheorem 4.4. Let Mn c Fn+P(c) be a compact and connected submanifold with parallel mean curvature, p > l,n> 4,c + H2 > 0. If the Ricci curvature of Mn is not less than (n - 2)c - 7-^H2 + |[(81n2 + 144n)/f4 + 36ncH2]^, then Mn is totally umbilical.