Euclidean hypersurfaces isometric to spheres

Given an immersed hypersurface Mn n in the Euclidean space E n +1 , the tangential component omega of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function sigma on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface Mn n in E n +1 of positive Ricci curvature with shape operator T invariant under omega and the support function sigma satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface Mn n in E n +1 with the gradient of support function sigma , an eigenvector of the shape operator T with eigenvalue function the mean curvature H , and the integral of the squared length of the gradient del sigma sigma has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface Mn n of positive Ricci curvature in E n +1 has an incompressible basic vector field omega , if and only if Mn n is isometric to a sphere.