Euclidean hypersurfaces isometric to spheres

Given an immersed hypersurface $M^n$ in the Euclidean space $E^{n+1}$, the tangential component $\mathit{\omega }$ 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 $M^n$ in $E^{n+1}$ of positive Ricci curvature with shape operator $T$ invariant under $\mathit{\omega }$ 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 $M^n$ 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 $\mathrm{\nabla }\sigma$ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $M^n$ of positive Ricci curvature in $E^{n+1}$ has an incompressible basic vector field $\mathit{\omega }$, if and only if $M^n$ is isometric to a sphere.