Summary: Let \(M\) be a compact Minimal hypersurface of the unit sphere \(S^{n+1}\). In this paper we use a constant vector field on \(\mathbb{R}^{n+2}\) to characterize the Clifford hypersurfaces \(S^l\big(\sqrt{\frac l n}\big) \times S^m \big(\sqrt{\frac m n}\big)\), \(l+m = n\), in \(S^{n+1}\). We also study compact minimal Einstein hypersurfaces of dimension greater than two in the unit sphere and obtain a lower bound for first nonzero eigenvalue 1 of its Laplacian operator.