Characterizations of Euclidean spheres

Abstract

<abstract> We use the tangential component $ \psi ^{T} $ of an immersion of a compact hypersurface of the Euclidean space $ \mathbf{E}^{m+1} $ in finding two characterizations of a sphere. In first characterization, we use $ \psi ^{T} $ as a geodesic vector field (vector field with all its trajectories geodesics) and in the second characterization, we use $ \psi ^{T} $ to annihilate the de-Rham Laplace operator on the hypersurface. </abstract>