Euclidean hypersurfaces isometric to spheres

Abstract

Given an immersed hypersurface $ M^{n} $ in the Euclidean space $ E^{n+1} $, the tangential component $\boldsymbol{\omega }$ 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 $\boldsymbol{\omega }$ 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 $ \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 $\boldsymbol{\omega }$, if and only if $ M^{n} $ is isometric to a sphere.

Keywords

• shape operator
• $ n $-sphere
• euclidean space
• static perfect fluid equation
• incompressible vector fields