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.

Given a conformal vector field $X$ defined on an $n$-dimensional Riemannian manifold $\left(N^n,g\right)$, naturally associated to $X$ are the conformal factor $\sigma$, a smooth function defined on $N^n$, and a skew symmetric $(1,1)$ tensor field $\mathrm{\Omega }$, called the associated tensor, that is defined using the $1$-form dual to $X$. In this article, we prove two results. In the first result, we show that if an $n$-dimensional compact and connected Riemannian manifold $\left(N^n,g\right)$, $n>1$, of positive Ricci curvature admits a nontrivial (non-Killing) conformal vector field $X$ with conformal factor $\sigma$ such that its Ricci operator $Rc$ and scalar curvature $\tau$ satisfy

$$Rc\left(X\right)=-(n-1)\mathrm{\nabla }\sigma \text{and}X(\tau )=2\sigma \left(n(n-1)c-\tau \right)$$

for a constant $c$, necessarily $c>0$ and $\left(N^n,g\right)$ is isometric to the sphere $S_c^n$ of constant curvature $c$. The converse is also shown to be true. In the second result, it is shown that an $n$-dimensional complete and connected Riemannian manifold $\left(N^n,g\right)$, $n>1$, admits a nontrivial conformal vector field $X$ with conformal factor $\sigma$ and associated tensor $\mathrm{\Omega }$ satisfying

$$Rc\left(X\right)=-div\mathrm{\Omega }\text{and}\mathrm{\Omega }\left(X\right)=0,$$

if and only if $\left(N^n,g\right)$ is isometric to the Euclidean space $\left(E^n,⟨,⟩\right)$.

Goddard's conjecture asserts that within de Sitter space, a spacelike hypersurface that is complete and has constant mean curvature must exhibit total umbilicity. Although counterexamples show that the conjecture does not hold in full generality, it remains valid under additional geometric assumptions, notably in the compact case. In this paper, we establish a broad extension of Goddard's conjecture to compact, spacelike hypersurfaces in Lorentzian manifolds that admit a timelike conformal vector field. Under a natural integral condition involving the ambient Ricci tensor and a mild assumption of the behavior of the mean curvature along the induced tangential flow, we prove that the hypersurface must be totally umbilical. As an application of our approach, we establish rigidity phenomena in Einstein Lorentzian manifolds and retrieve, as a special instance, the well-known classification of compact, spacelike hypersurfaces with constant mean curvature in de Sitter space as round spheres.