Hypersurfaces in a Euclidean space with a Killing vector field

An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space ${\mathbb{R}}^{n+1}$, $n>1$ admits a unit Killing vector field $\xi$ that leaves the shape operator $S$ invariant and has sectional curvatures of plane sections containing $\xi$ positive which satisfies $S(\xi )=\alpha \xi$, $\alpha$ mean curvature if, and only if, $n=2m-1$, $\alpha$ is constant and the hypersurface is isometric to the sphere $S^{2m-1}({\alpha }^2)$. Similarly, we found another characterization of the unit sphere $S^2({\alpha }^2)$ using the smooth function $\sigma =g(S(\xi ),\xi )$ on the hypersurface.