Rigidity of hypersurfaces in a Euclidean sphere

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an n(>= 2)-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss-Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean n-sphere. We also show that an n(>= 2)-dimensional complete connected orientable hypersurface immersed in a unit sphere Sn+1 whose Gauss image is contained in a closed geodesic ball of radius less than pi/2 in Sn+1 is diffeomorphic to a sphere. Finally, we prove that an n(>= 2)-dimensional connected closed orientable hypersurface in Sn+1 with constant scalar curvature greater than n(n - 1) and Gauss image contained in an open hemisphere is totally umbilic.