Some generic hypersurfaces in a Euclidean space

In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field $ {\omega } $ on a compact and connected hypersurface $ N $ of the Euclidean space $ R^{m+1} $ with a mean curvature $ \alpha $ constant along the integral curves of $ {\omega } $ and a shape operator $ T $ satisfying $ T({\omega) = \alpha \omega } $ implies that $ \alpha $ is a constant and $ N $ is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field $ \mathbf{v} $ on a compact and connected hypersurface $ N $ of a Euclidean space $ R^{m+1} $ gives a nonzero function $ \sigma = g\left(T \mathbf{v}, \mathbf{v}\right) $ with shape operator $ T $, and the integral of the function $ m\alpha \sigma Ric\left(\mathbf{v}, \mathbf{v}\right) $ has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface $ N $ with support $ \rho $ and basic vector field $ \mathbf{u} $, the integral of the Ricci curvature $ Ric\left(\mathbf{u}, \mathbf{u}\right) $ has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.