Euclidean hypersurfaces isometric to spheres

Given an immersed hypersurface Mⁿ in the Euclidean space Eⁿ⁺¹, the tangential component ω 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 σ 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 in Eⁿ⁺¹ of positive Ricci curvature with shape operator T invariant under ω and the support function σ 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 in Eⁿ⁺¹ with the gradient of support function σ, an eigenvector of the shape operator T with eigenvalue function the mean curvature H, and the integral of the squared length of the gradient ∇ σ 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 of positive Ricci curvature in Eⁿ⁺¹ has an incompressible basic vector field ω, if and only if Mⁿ is isometric to a sphere.