Submanifolds With Constant Scalar Curvature And The Harmonic Function Of Finsler Manifold

This paper consists of two parts,three chapters totally. In the first part, which contains the first and the second chapter,we investgate the pinching problems of submanifolds in unit sphere S^n+p with constant scalar curvature and spacelike submanifolds in the de Sitter space S_p^n+p(c) with constant scalar curvature, and we obtains a rigidity theorem respectively. In the second part of the paper we study the harmonic function of Finsler manifold.Let Mⁿ be a connected and oriented submanifold isometrically immersed in a space form M_cⁿ⁺¹ (c≥0). We say Mⁿ is closed if it is compact and without boundary. Denote by R, H and S the normalized scalar curvature, the mean curvature and the square of the length of the second fundamental form of Mⁿ, respectively. The classifying problem of submanifolds is an interesting problem. There have been many rigidity results for minimal submanifolds and submanifolds with parallel mean curvature vector field immersed into a unit sphere([9],[12],[23],[25]). Naturally, the next studying case is that Mⁿ is a submanifold in a unit sphere S^n+p with constant scalar curvature. Cheng and Yau firstly studied the rigidity problem for hypersurface Mⁿ into a space form M_cⁿ⁺¹ (c≥0) with constant scalar curvature by introducing a self-adjoint second order differential operator([7]), which is still one of the most important tool to study submanifolds with constant scalar curvature. Because the condition is too weak to be studied, we always add another condition, for example, the section curvature has low bound or the normal bundle is fiat and so on([7],[11]). In the first chapter of the paper, we study the case that Mⁿ is a submanifold in a unit sphere S^n+p with constant normalized scalar curvature and parallel normalized mean curvature vector field. Clearly, hypersurfaces always satisfy the last condition. There have been some results on this aspect([11][13][33]). But [13] and [33] only study the case when Mⁿ is a hypersurface and the pinching results about the mean curvature in [11] isn't good (for the pinching results is a equation about R,S and H). In the first chapter of the paper, we study the pinching problem on S for submanifold Mⁿ with the same assumptions and get a theorem alike the...