The Geometry And Topology Of Submanifolds Of The Unit Sphere

In this thesis, we mainly study two kinds of problems about submanifolds of the unit spheres. One is an isometric problem about the compact hypersurface with constant scalar curvature, which has only two distinct principal curvatures one of which is simple. The other is the sphere theorem problems on orientable compact submanifolds.As to the first problem, Cheng Q.M. [2] put forward an open problem: let M be acomplete hypersurface with constant scalar curvature n(n - 1)r ( r =(n-2)/(n-1)) in the unitsphere , if M has only two distinct principal curvatures one of which is simple, then, is M isometric to ? The thesis presents a partially affirmativeanswer to an open problem and proves the conclusion is correct if the submanifold is compactness and orientable. Differing from the former methods, we make use of the mean curvature to prove that the sectional curvature is non-negative if the submanifold is compactness and orientable, and avoid discussing the differential equation. Furthermore, the author utilizes the method above to discuss the submanifolds with a general constant scalar curvature, and obtains some conclusions which are analogous to the ones of Cheng Q.M. But the proof is simpler than the former.As to the second problem, the author proves two sphere theorems on the submainfolds of the unit sphere, giving two conditions which make the submanifolds homemorphic to the sphere, respectively. One condition is an inequality between Ricci curvature and the mean curvature in the even dimensions. The other is about the scalar curvature in the case of the minimal submanifolds. And the author points out the significations of the conclusions. The main results of the thesis have been accepted for publication by Journal of Anhui University.