Geometry Of Submanifolds In A Kind Of Product Spaces

Geometry of submanifolds is an important topic of differential geometry. In recent years, the submanifolds in product spaces were extensively studied. In the paper, we study submani-folds with parallel mean curvature vector and Willmore submanifolds in Mn(c) x R.Firstly, in Chapter 3, we study the submanifolds in products of semi-Riemannian space forms. In 2011, M. Batista[1] introduced a (1,1) tensor S on the surface ?2(?)M2(c) x R. Later on, Fetcu and H. Rosebberg[2] generalized the operator S to the surface ?2(?)Mn(c) x R. We consider a more general tensor S on an immersed surface ?2(?) (Mn(c) × R, g?) and study the gap phenomena of[S], then we obtain some Pinching constants. Especially for the case of Riemmannian product spaces M2(c) x R, these Pinching constants are bigger than the corresponding in [1].Secondly, we consider the surfaces with non-negative Gaussian curvature in Mn(c) x R, and under the additional condition of constant angle, we characterize flat surfaces in Mn(c) x R. It is just to solve an open problem proposed by H. Alencar, M. do Carmo and R. Tribuzy in [3]. As known to all, it is very difficult to completely classify flat surfaces of Mn(c) x R, even for the surfaces ?2(?) M2(c) x R. We obtain the parameter representation of flat surfaces in Mn(c) x R under the additional condition of constant angle.Thirdly, in Chapter 5, we study the rigidity of PMC submanifolds in M"(c)xR. By comput-ing the Laplace of operators, we obtain several Simons type equations. Then we get several gap theorems by working on these Simons type equations. In detail, for submanifolds in Sn(1) x R, we prove that the submanifolds which meet some conditions are totally geodesic in Sn(1); for surfaces in M3(c)×R, we obtain several rigidity theorems and theorem 5.14 improves Proposition 4.1 of [4] under additional conditions; for submanifolds in M"(c) x R, under certain conditions, we proof that the submanifolds is a totally umbilical CMC hypersurface in Mn+1(c) of Mm(c).Finally, in Chapter 6, the Willmore submanifolds in Mn(c) x R are studied. By calcu-lating the variation of the functional Fk(x) (k= n/2, it is Willmore functional), we get the Euler-Lagrange equation. Then we give the necessary and sufficient condition of Willmore sub-manifolds. We prove that Willmore surfaces with property of constant angle in M2(c) x R is nothing but ?2 c M2(c) or ?2=?×R, where y is a curve of M2(c). And a totally umbilical surface immersed in M2(c) × R must be a Willmore surface. But the inverse proposition maybe not be true. Then, we give a sufficient condition that insure the inverse proposition true.