Results Of The Submanifolds In The Space Form With Constant Curvature

The aim of this paper is to deal with the submanifolds immersed into the space form with constant curvature and obtain the following results:1. For the compact submanifolds of the standard Euclidean sphere, we shall give a Simons-type inequality involving the squared norm S of the second fundamental form h of Mⁿ in terms of the first nonzero eigenvalueλ₁ of the Laplacian of Mⁿ. Furthermore, if, in addition, S is a constant, we give the lower bound for S.2. For the compact submanifolds of the Euclidean space, we shall prove the non-existence of stable currents for certain classes of them. For hypersurfaces, we prove the non-existence theorems under the assumptions about the principal curvature, sectional curvature or the square length of the second fundamental form respectively. For high-codimension, we also prove that there are no stable currents in submanifolds of the Euclidean space when the square length of the second fundamental form satisfies a pinching condition. As a result, such submanifolds are homeomorphic to the Euclidean sphere.3. For the submanifolds of the hyperbolic space, we consider the hypersurfaces with non-positive Ricci curvature and give a lower bound for the square length of the second fundamental form of the hypersurfaces. Further, we obtain an upper bound for the product of the principal curvatures of the hypersurfaces.