New Characterizations of Hyperspheres and Spherical Hypercylinders in Euclidean Space

Let x be an isometric immersion of a Riemannian n-manifold M into a Euclidean n+1-space En+1 which does not pass through the origin of En+1. Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector field, support function, and mean curvature, we establish three new characterizations of hyperspheres. In addition, we prove that if the energy function of M satisfies the static perfect fluid equation, then M has at most two distinct principal curvatures. As an application, we prove that a complete noncompact hypersurface M is a spherical hypercylinder if the energy function of M satisfies the static perfect fluid equation, and it has exactly two distinct nonsimple principal curvatures.