: In this thesis, we present an expanded account of a characterization of sphere in Euclidean space and spherical submanifolds in a Euclidean space based on two article by H. Aloda and S. Deshmukh(2007). Given an n-dimensional submanifold M of a Euclidean space with immersion : M , the position vector field plays an important role in studying the geometry of the submanifolds. For an orientable compact and connected hypersurface in the Euclidean space R with positive scalar curvature S , the shape operator A and the mean curvature , it is shown that the inequality implies that the hypersurface is a sphere, where is the gradient of . A similar characterization is also obtained for spheres the Euclidean space .
