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.